#6447998

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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)(9,10,13,15)(11,14,12,16)(17,18,21,22)(19,23,27,26)(20,24,28,29)(25,30,32,31);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(18,22)(19,23)(20,24)(26,27)(28,29)(30,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19,21,27)(18,23,22,26)(20,25,28,32)(24,30,29,31);(1,4,5,3)(2,8,6,7)(17,20,21,28)(18,24,22,29)(19,25,27,32)(23,30,26,31);(17,18,21,22)(19,26,27,23)(20,24,28,29)(25,31,32,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=f,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, 6355194);