#6628946

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4633 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536142);