#6874862

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 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)(3,7,8,5)(17,18,22,26,23,27,21,35)(19,28,30,40,31,39,37,24)(20,32,33,41,34,25,36,42)(29,44,46,38,47,43,45,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14);(3,8)(5,7)(10,14)(15,16)(17,19,23,31)(18,24,27,40)(20,29,34,47)(21,30,22,37)(25,38,32,48)(26,39,35,28)(33,45,36,46)(41,43,42,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,38)(26,42)(27,32)(28,43)(30,46)(31,47)(35,41)(37,45)(39,44)(40,48);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,27,35)(19,30,31,37)(20,33,34,36)(24,39,40,28)(25,41,32,42)(29,45,47,46)(38,43,48,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,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, 6782058);