#6442762

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4696 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)(11,16)(12,24)(13,18)(14,28)(15,19)(17,26)(20,33)(21,30)(22,31)(23,37)(25,32)(27,38)(29,36)(34,39)(35,40);(2,4)(5,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,29)(18,30)(19,31)(24,33)(25,34)(26,36)(27,35)(28,37)(32,39)(38,40);(6,8)(9,12,13,25,15,27,14,26)(10,17,28,38,19,32,18,24)(11,20,21,34,22,35,23,36)(16,29,37,40,31,39,30,33);(6,8)(9,13,15,14)(10,18,19,28)(11,21,22,23)(12,25,27,26)(16,30,31,37)(17,24,32,38)(20,34,35,36)(29,33,39,40);(11,22)(16,31)(20,35)(21,23)(29,39)(30,37)(33,40)(34,36)

Polycyclic presentation:

2^9|c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315555, #72315565, #72315569
maximal normal subgroup of	#72315555, #72315565, #72315569
maximal quotient of	#72315782, #72315825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349958);