#513185

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33849	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,4)(5,6,13,8)(7,19,24,56)(9,30,33,51)(10,34,39,49)(11,40,43,65)(12,29,42,18)(14,44,25,21)(15,38,26,63)(16,32,27,37)(17,52,28,36)(20,55,23,48)(22,60,61,54)(31,47,59,45)(35,53,68,66)(41,50,57,64)(46,58,62,67);(5,7,13,24)(6,14,8,25)(9,31,33,59)(10,35,39,68)(11,41,43,57)(12,20,42,23)(15,48,26,55)(16,50,27,64)(17,53,28,66)(18,45,29,47)(19,49,56,34)(21,32,44,37)(22,52,61,36)(30,60,51,54)(38,58,63,67)(40,46,65,62);(2,4)(5,8,13,6)(7,14,24,25)(9,26,33,15)(10,27,39,16)(11,28,43,17)(12,29,42,18)(19,44,56,21)(20,45,23,47)(22,46,61,62)(30,63,51,38)(31,48,59,55)(32,34,37,49)(35,50,68,64)(36,65,52,40)(41,53,57,66)(54,67,60,58);(5,9)(6,15)(7,20)(8,26)(10,36)(11,38)(12,37)(13,33)(14,45)(16,40)(17,51)(18,34)(19,54)(21,58)(23,24)(25,47)(27,65)(28,30)(29,49)(31,35)(32,42)(39,52)(41,57)(43,63)(44,67)(48,50)(53,66)(55,64)(56,60)(59,68)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420381);