#495339

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,g^b=gi,d^c=di

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, 402535);