#6629600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628847
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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)(17,18,22,19)(20,28,30,32)(21,26,23,25)(24,31,27,29);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,22,18)(20,27,30,24)(21,26,23,25)(28,29,32,31);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,19)(24,27)(25,26)(28,32);(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(17,21,22,23)(18,25,19,26)(20,29,30,31)(24,28,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536796);