#502882

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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,d^c=di

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, 410078);