#6746173

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6653369);