#6444284

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

2756 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)(12,25)(13,16)(14,18)(15,19)(17,30)(20,26)(22,37)(23,34)(24,32)(27,40)(28,33)(29,35)(31,39)(36,38);(1,2)(3,4)(6,7)(9,11)(10,16)(12,22)(13,15)(14,24)(17,31)(18,34)(19,21)(20,32)(23,26)(25,40)(27,29)(28,39)(30,38)(33,36)(35,37);(2,4)(5,6,8,7)(9,12,26,28,15,29,14,30)(10,17,18,35,19,33,20,25)(11,22,23,39,13,27,24,38)(16,31,34,37,21,36,32,40);(2,4)(5,7,8,6)(9,13,15,11)(10,16,19,21)(12,27,29,22)(14,23,26,24)(17,31,33,36)(18,34,20,32)(25,40,35,37)(28,38,30,39);(9,14,15,26)(10,18,19,20)(11,23,13,24)(12,28,29,30)(16,32,21,34)(17,25,33,35)(22,38,27,39)(31,37,36,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,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, 6351480);