#503128

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1245 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)(12,34)(13,20)(14,27)(16,25)(19,46)(21,52)(22,38)(24,57)(26,54)(28,55)(29,61)(31,43)(32,37)(33,56)(35,65)(36,63)(39,68)(40,60)(41,59)(42,64)(44,49)(45,47)(48,66)(50,53)(51,62)(58,67);(1,2,4,3)(5,7,9,20)(6,13,15,18)(8,24,26,50)(10,21,29,51)(11,32,12,22)(14,39,40,67)(16,36,43,45)(17,38,34,37)(19,33,48,28)(23,53,54,57)(25,47,31,63)(27,58,60,68)(30,62,61,52)(35,42,44,41)(46,55,66,56)(49,64,65,59);(5,8,25,59,12,28,29,60,9,26,31,64,11,33,10,27)(6,14,30,56,17,42,43,54,15,40,61,55,34,41,16,23)(7,19,47,68,22,50,51,65,20,48,63,58,32,24,21,49)(13,35,62,53,38,39,45,46,18,44,52,57,37,67,36,66);(1,3,4,2)(6,15)(13,18)(14,40)(16,43)(17,34)(23,54)(30,61)(35,44)(36,45)(37,38)(39,67)(41,42)(46,66)(52,62)(53,57)(55,56)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410324);