#356342

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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

1046 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:

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

Polycyclic presentation:

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

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, 263538);