#7090815

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,25,21,24,28,22)(19,23,26,30,27,31,32,29);(9,10)(11,15)(12,14)(13,16)(18,22)(19,26)(20,28)(24,25)(27,32)(29,30);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15);(11,16)(13,15)(17,19,20,26,21,27,28,32)(18,23,25,30,24,31,22,29);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,24)(19,27)(22,25)(26,32)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6998011);