#6858928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#30414	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 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)(19,25)(20,34)(21,27)(22,28)(23,38)(24,29)(26,35)(30,40)(31,41)(32,47)(33,42)(36,44)(37,43)(39,45)(46,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,33)(26,39)(27,44)(28,41)(29,42)(34,40)(35,45)(37,46)(38,47)(43,48);(10,14)(15,16)(17,20,24,21)(18,26,29,43)(19,30,33,36)(22,35,23,37)(25,39,42,48)(27,38,34,28)(31,45,32,46)(40,41,44,47);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,27,29,34)(19,30,33,36)(22,35,23,37)(25,40,42,44)(26,38,43,28)(31,46,32,45)(39,41,48,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,28,29,38)(19,31,33,32)(20,35,21,37)(25,41,42,47)(26,27,43,34)(30,45,36,46)(39,44,48,40)

Polycyclic presentation:

2^9|c^2=i,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6766124);