#6370875

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

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

4318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,h^c=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278071);