#6444286

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2742 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351482);