#502092

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409288);