#6437889

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×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4682 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,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, 6345085);