#7606925

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1487 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,23,25)(19,26,21,22)(20,24,28,31)(27,32,29,30);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,18,23,21,26,25)(20,27,30,24,28,29,32,31);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,31)(26,32);(2,6)(5,7)(11,16)(14,15)(20,28)(24,31)(27,29)(30,32);(2,6)(5,7)(17,21,22,25,23,19,26,18)(20,29,30,31,28,27,32,24)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=h,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7514121);