#6458707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,24,29)(19,20,26,32)(21,27,25,31)(22,28,23,30);(9,11)(10,13)(12,15)(14,16)(17,19,23,27,24,26,22,31)(18,25,28,20,29,21,30,32);(2,5)(6,8)(11,15)(13,16)(17,20,24,32)(18,26,29,19)(21,22,25,23)(27,28,31,30);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21)(18,27)(19,30)(20,22)(23,32)(24,25)(26,28)(29,31);(17,22,24,23)(18,28,29,30)(19,31,26,27)(20,21,32,25)

Polycyclic presentation:

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

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, 6365903);