#6431970

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4625 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339166);