#6438818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3234 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,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, 6346014);