#512766

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	C₂²×D₃₂	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 454 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,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, #72321360, #72321597, …
maximal normal subgroup of	#72232007, #72321360, #72321597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419962);