#7620210

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1248 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)(11,16)(12,15)(13,14)(17,18,23,22)(19,24,28,29)(20,25,31,30)(21,27,32,26);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,27)(21,30)(22,29)(23,28)(25,32)(26,31);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,23,31)(18,25,22,30)(19,27,28,26)(21,24,32,29);(9,11,13,12)(10,15,14,16)(17,21,18,26,23,32,22,27)(19,20,29,25,28,31,24,30);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,14,16)(17,18,23,22)(19,24,28,29)(20,30,31,25)(21,26,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7527406);