#7079428

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	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,24,28,22,31)(19,29,21,26,25,32,30,20);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,20)(21,22)(23,30)(24,25)(26,28)(27,32)(29,31);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,31)(22,32)(23,29)(24,26)(27,30);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,24,30,23,25)(18,26,31,29,28,20,27,32);(10,14)(13,15)(17,22,24,23)(18,27,28,31)(19,30,25,21)(20,29,26,32)

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,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, 6986624);