#368206

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1432 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275402);