#6628856

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#585	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

17018 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)(5,6)(7,8)(9,10,15,19)(11,20,23,30)(12,17,25,33)(13,18,14,27)(16,28,31,22)(21,34,36,39)(24,32,26,37)(29,38,40,35);(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,35)(25,36)(26,38)(32,39)(33,40)(34,37);(2,4)(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,29)(18,32)(19,33)(20,34)(22,35)(23,36)(27,37)(28,38)(30,39)(31,40);(6,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,25,26)(16,30,31,20)(17,32,33,37)(21,35,36,38)(29,39,40,34);(12,25)(17,33)(21,36)(24,26)(29,40)(32,37)(34,39)(35,38)

Polycyclic presentation:

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

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, 6536052);