#7016570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42906	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6923766);