#7254843

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7162039);