#6442538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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

4636 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)(11,19)(12,17)(13,25)(14,16)(15,21)(18,27)(20,33)(22,32)(23,30)(24,37)(26,28)(29,35)(31,36)(34,40)(38,39);(5,6)(7,8)(9,11,14,22)(10,15,16,28)(12,20,23,34)(13,26,27,21)(17,29,30,39)(18,32,25,19)(24,38,36,35)(31,40,37,33);(1,2,3,4)(6,8)(10,16)(15,28)(17,30)(18,25)(19,32)(29,39)(31,37)(33,40);(6,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,29)(16,30)(18,31)(19,33)(21,35)(22,34)(25,37)(26,38)(27,36)(28,39)(32,40);(12,23)(17,30)(20,34)(24,36)(29,39)(31,37)(33,40)(35,38)

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^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, 6349734);