#7080730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#23866	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6987926);