#6436132

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=g,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343328);