#7007875

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(17,18,22,28,23,27,19,24)(20,30,29,21,26,32,31,25);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,28)(20,31,26,29)(21,30,25,32);(9,11)(10,13)(12,16)(14,15)(17,20,23,26)(18,25,27,21)(19,29,22,31)(24,30,28,32);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,21)(18,26)(19,30)(20,27)(22,32)(23,25)(24,29)(28,31);(10,14)(13,15)(18,27)(21,25)(24,28)(30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6915071);