#6372235

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342612
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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=gi,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=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, 6279431);