#6372684

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,e^c=ei,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, 6279880);