#6793516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 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)(4,8)(5,7)(10,13)(11,15)(17,18,23,29)(19,31,34,25)(20,27,21,36)(22,30,24,28)(26,38,43,35)(32,41,37,44)(33,42,39,45)(40,48,46,47);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19)(18,25)(20,32)(21,37)(22,39)(23,34)(24,33)(26,40)(27,44)(28,45)(29,31)(30,42)(35,48)(36,41)(38,47)(43,46);(10,13)(14,16)(17,20,23,21)(18,26,29,43)(19,32,34,37)(22,35,24,38)(25,40,31,46)(27,30,36,28)(33,47,39,48)(41,45,44,42);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,27,29,36)(19,32,34,37)(22,35,24,38)(25,41,31,44)(26,30,43,28)(33,48,39,47)(40,45,46,42);(1,5,6,4)(2,7,3,8)(17,22,23,24)(18,28,29,30)(19,33,34,39)(20,35,21,38)(25,42,31,45)(26,27,43,36)(32,47,37,48)(40,41,46,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6700712);