#6372667

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4622 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)(5,6,8,7)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,21)(22,35)(23,29)(24,32)(25,33)(26,34)(28,31)(30,36)(37,40)(38,39);(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,36)(25,37)(26,38)(32,39)(33,35)(34,40);(2,4)(9,12,15,14)(10,17,19,27)(11,21,23,28)(13,24,26,25)(16,29,31,20)(18,32,34,33)(22,36,38,37)(30,39,40,35);(5,7,8,6)(9,13,15,26)(10,18,19,34)(11,22,23,38)(12,24,14,25)(16,30,31,40)(17,32,27,33)(20,35,29,39)(21,36,28,37);(9,12,15,14)(10,17,19,27)(11,21,23,28)(13,25,26,24)(16,29,31,20)(18,33,34,32)(22,37,38,36)(30,35,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279863);