#6461502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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	#23428	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2118 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)(17,18)(19,29)(20,26)(21,28)(22,27)(23,25)(24,30)(31,32);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,25,32)(21,30,22,31)(26,27,29,28);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(18,25)(24,32)(26,29)(27,28);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,23,19)(18,26,25,29)(21,30,22,31)(24,27,32,28);(17,21,23,22)(18,27,25,28)(19,30,20,31)(24,29,32,26)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6368698);