#7094750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1292 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7001946);