#500675

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60649	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1154 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:

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

Polycyclic presentation:

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

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, 407871);