#6874879

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(17,18,22,25,21,27,20,26)(19,23,28,32,30,24,29,31);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,32)(26,31);(3,8)(5,7)(10,14)(15,16)(17,18,21,27)(19,24,30,23)(20,25,22,26)(28,32,29,31);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32);(9,12,13,11)(10,15,14,16)(17,20,21,22)(18,25,27,26)(19,28,30,29)(23,31,24,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=i,d^2=fhi,e^2=fh,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, 6782075);