#6437885

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3497	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4619 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)(9,10,13,27,15,20,14,19)(11,18,28,16,23,31,22,32)(12,24,29,17,26,34,25,35)(21,30,36,33,38,40,37,39);(2,4)(5,6)(7,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,30)(18,27)(19,31)(20,32)(24,33)(25,36)(26,38)(29,37)(34,39)(35,40);(6,8)(9,12,15,26)(10,17,20,35)(11,21,23,38)(13,25,14,29)(16,30,32,40)(18,33,31,39)(19,34,27,24)(22,36,28,37);(6,8)(9,11)(10,18)(12,21)(13,28)(14,22)(15,23)(16,27)(17,33)(19,32)(20,31)(24,30)(25,37)(26,38)(29,36)(34,40)(35,39);(9,13,15,14)(10,19,20,27)(11,22,23,28)(12,25,26,29)(16,31,32,18)(17,34,35,24)(21,36,38,37)(30,39,40,33)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6345081);