#7198385

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

1422 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)(10,14)(12,15)(18,24)(19,26)(21,23)(25,32)(27,28)(29,30);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,16,15)(17,18)(19,27)(20,23)(21,31)(22,24)(25,30)(26,32)(28,29);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,19,23,26,22,30,21,29)(18,25,20,28,24,27,31,32);(3,8)(6,7)(10,14)(11,16)(17,20)(18,21)(19,28)(22,31)(23,24)(25,29)(26,27)(30,32);(9,12,13,15)(10,11,14,16)(17,21,22,23)(18,20,24,31)(19,29,30,26)(25,28,27,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fgh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=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, 7105581);