#6442547

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4642 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349743);