#7089665

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(18,24)(19,28)(21,22)(25,32)(26,27)(29,30);(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,30)(21,24)(22,26)(23,27)(25,28)(31,32);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,22)(21,31)(23,28)(24,30)(26,32)(27,29);(11,16)(13,15)(17,20,23,31)(18,25,27,29)(19,21,28,22)(24,32,26,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,26,27,24)(19,20,28,31)(25,30,29,32)

Polycyclic presentation:

2^9|c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6996861);