#6432443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339639);