#7610203

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1492 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,29,28)(20,24,30,32)(21,27,31,25);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,26,28)(20,27,30,25)(21,24,31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,22,30)(18,24,26,32)(19,27,29,25)(21,23,31,28);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,26,27,22,31,18,25)(19,20,23,32,29,30,28,24);(2,6)(7,8)(20,30)(21,31)(24,32)(25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7517399);