#495504

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	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 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)(3,6)(4,5)(7,8)(9,10,13,18,14,19,26,32,15,11,16,21,27,22,29,35)(12,17,24,30,25,31,37,39,23,20,28,33,36,34,38,40);(9,11,13,18,14,22,26,32,15,10,16,21,27,19,29,35)(12,20,24,30,25,34,37,39,23,17,28,33,36,31,38,40);(2,5)(6,8)(9,12)(10,17)(11,20)(13,24)(14,25)(15,23)(16,28)(18,30)(19,31)(21,33)(22,34)(26,37)(27,36)(29,38)(32,39)(35,40);(1,3,4,7)(2,6,5,8)(12,23)(17,20)(24,28)(25,36)(30,33)(31,34)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^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, 402700);