#6450711

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6357907);