#6458355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,24,22,27,21,26)(19,28,30,32,20,25,29,31);(2,6)(7,8)(18,24)(19,29)(20,30)(21,23)(26,27)(31,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,21,29,22,20,23,30)(18,25,26,32,27,28,24,31);(1,4,5,3)(2,8,6,7)(17,20,21,30,22,19,23,29)(18,25,26,32,27,28,24,31);(19,20)(25,28)(29,30)(31,32)

Polycyclic presentation:

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

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, 6365551);