#6371012

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4014 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)(9,10,13,18,14,17,26,31)(11,19,21,32,22,15,25,28)(12,16,24,30,23,29,36,39)(20,33,35,40,34,27,37,38);(2,5)(6,8);(9,11,14,22)(10,15,17,19)(12,20,23,34)(13,25,26,21)(16,27,29,33)(18,32,31,28)(24,37,36,35)(30,40,39,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,29)(18,30)(19,33)(21,35)(22,34)(25,37)(26,36)(28,38)(31,39)(32,40);(12,23)(16,29)(20,34)(24,36)(27,33)(30,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=f,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, 6278208);