#502877

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,16,22)(11,17,26,42)(12,27,33,60)(13,34,37,49)(14,21,15,20)(18,43,47,30)(19,36,50,29)(23,52,55,72)(24,56,35,67)(25,51,38,41)(28,57,62,70)(31,59,32,58)(39,68,69,54)(40,63,48,53)(44,61,71,65)(45,66,46,64);(2,5)(6,7)(9,11)(10,17)(12,28)(13,35)(14,38)(15,25)(16,26)(18,44)(19,48)(20,51)(21,41)(22,42)(23,31)(24,37)(27,57)(29,63)(30,65)(32,55)(33,62)(34,67)(36,53)(39,45)(40,50)(43,61)(46,69)(47,71)(49,56)(52,59)(54,64)(58,72)(60,70)(66,68);(9,12,29,64,15,32,37,43,16,33,36,66,14,31,13,30)(10,18,34,59,21,46,50,60,22,47,49,58,20,45,19,27)(11,23,53,61,25,28,35,68,26,55,63,65,38,62,24,54)(17,39,56,70,41,44,48,72,42,69,67,57,51,71,40,52);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=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, 410073);