#6441936

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1936 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6349132);