#357000

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=fgi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264196);