#6969794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 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)(3,6)(4,8)(5,7)(10,14)(11,12)(17,18,21,26,22,25,32,24)(19,20,29,30,28,23,27,31);(1,2,6,3)(4,7,5,8)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,25)(21,29)(22,28)(24,30)(26,31)(27,32);(1,3,6,2)(4,7,5,8)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,19,25,28)(21,31,32,30)(24,29,26,27);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,15,14,16)(17,21,22,32)(18,24,25,26)(19,27,28,29)(20,30,23,31);(1,5,6,4)(2,7,3,8)(18,25)(19,28)(24,26)(27,29)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6876990);