#6458690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,23,26)(19,21,25,22)(20,28,32,24)(27,30,29,31);(9,11)(10,13)(12,15)(14,16)(17,19,22,18,23,25,21,26)(20,29,30,28,32,27,31,24);(2,5)(6,8)(11,15)(13,16)(17,20,23,32)(18,24,26,28)(19,27,25,29)(21,31,22,30);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,21,23,22)(18,25,26,19)(20,30,32,31)(24,27,28,29);(17,21,23,22)(18,25,26,19)(20,31,32,30)(24,29,28,27)

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=dh,e^b=ei,d^c=di

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, 6365886);