#7325696

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,21,23)(19,27,30,42)(20,25,32,37)(22,35,28,26)(24,38,36,31)(29,43,33,46)(34,48,44,41)(39,47,40,45);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(24,36)(26,35)(27,42)(29,33)(34,44)(45,47);(9,11)(10,13)(12,16)(14,15)(17,19,28,38,21,30,22,31)(18,24,35,42,23,36,26,27)(20,33,44,45,32,29,34,47)(25,40,48,43,37,39,41,46);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,29)(21,32)(22,34)(23,37)(24,39)(26,41)(27,43)(28,44)(30,33)(31,45)(35,48)(36,40)(38,47)(42,46);(2,6)(5,7)(10,14)(13,15)(18,23)(19,30)(20,32)(26,35)(31,38)(34,44)(39,40)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei,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, 7232892);