#6791882

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6699078);