#6458403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

1724 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)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,23,26)(19,21,25,22)(20,24,30,32)(27,29,28,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,18,23,25,21,26)(20,28,31,32,30,27,29,24);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,31)(23,30)(25,28)(26,32);(11,15)(13,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,28,32,27);(20,30)(24,32)(27,28)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6365599);