#512770

order	512
nilpotency class	5
isoclinism	isoclinic class of #33982
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,4,3)(5,6,11,18)(7,21,8,15)(9,20,13,16)(10,30,33,55)(12,37,36,47)(14,26,40,23)(17,48,50,34)(19,29,52,32)(22,42,25,45)(24,57,27,61)(28,62,35,65)(31,53,60,43)(38,49,59,44)(39,56,54,58)(41,67,68,63)(46,64,51,66);(5,7,11,8)(6,14,18,40)(9,22,13,25)(10,31,33,60)(12,38,36,59)(15,30,21,55)(16,39,20,54)(17,49,50,44)(19,53,52,43)(23,48,26,34)(24,29,27,32)(28,63,35,67)(37,41,47,68)(42,65,45,62)(46,57,51,61)(56,66,58,64);(1,3,4,2)(5,8,11,7)(6,15,18,21)(9,22,13,25)(10,26,33,23)(12,27,36,24)(14,30,40,55)(16,45,20,42)(17,43,50,53)(19,44,52,49)(28,56,35,58)(29,38,32,59)(31,34,60,48)(37,61,47,57)(39,62,54,65)(41,46,68,51)(63,66,67,64);(5,9,11,13)(6,16,18,20)(7,22,8,25)(10,28,33,35)(12,29,36,32)(14,39,40,54)(15,42,21,45)(17,46,50,51)(19,47,52,37)(23,56,26,58)(24,38,27,59)(30,65,55,62)(31,63,60,67)(34,64,48,66)(41,53,68,43)(44,61,49,57)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232007, #72321369, #72321615, …
maximal normal subgroup of	#72232007, #72321369, #72321615, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419966);