#7072981

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#44996	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6980177);