#7034428

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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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)(17,18,23,29)(19,24,33,41)(20,26,35,43)(21,28,22,27)(25,36,42,34)(30,38,45,48)(31,40,32,39)(37,46,47,44);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,33)(18,24,29,41)(20,30,35,45)(21,31,22,32)(25,37,42,47)(26,38,43,48)(27,39,28,40)(34,44,36,46);(10,13)(11,16)(17,20,23,35)(18,25,29,42)(19,30,33,45)(21,34,22,36)(24,37,41,47)(26,28,43,27)(31,44,32,46)(38,40,48,39);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20)(18,26)(19,30)(21,36)(22,34)(23,35)(24,38)(25,27)(28,42)(29,43)(31,46)(32,44)(33,45)(37,39)(40,47)(41,48);(10,13)(14,15)(17,21,23,22)(18,27,29,28)(19,31,33,32)(20,34,35,36)(24,39,41,40)(25,26,42,43)(30,44,45,46)(37,38,47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6941624);