#7034459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1549 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,28)(29,38)(31,40)(32,43)(33,37)(34,41)(35,46)(39,42)(44,48)(45,47);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,29,34,35)(25,30,40,37)(26,38,41,46)(32,44,45,42)(39,43,48,47);(10,13)(11,16)(17,20,23,33)(18,25,27,40)(19,28,22,31)(21,32,34,45)(24,37,36,30)(26,39,41,48)(29,42,35,44)(38,47,46,43);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,29)(20,32)(22,35)(23,34)(24,38)(25,39)(27,41)(28,42)(30,43)(31,44)(33,45)(36,46)(37,47)(40,48);(10,13)(14,15)(18,27)(21,34)(24,36)(25,40)(29,35)(30,37)(32,45)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941655);