#7020735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6927931);