#6371015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,21,15,20,13,19)(11,16,24,33,25,32,23,31)(12,18,26,35,27,17,28,34)(22,30,36,40,37,29,38,39);(2,5)(6,8)(9,11,15,25)(10,16,20,32)(12,22,27,37)(13,23,14,24)(17,29,18,30)(19,31,21,33)(26,36,28,38)(34,39,35,40);(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,29)(18,20)(19,34)(21,35)(23,36)(24,38)(25,37)(30,32)(31,39)(33,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,30)(17,20)(19,34)(21,35)(23,38)(24,36)(25,37)(29,32)(31,39)(33,40);(9,13,15,14)(10,19,20,21)(11,23,25,24)(12,26,27,28)(16,31,32,33)(17,34,18,35)(22,36,37,38)(29,39,30,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278211);