#368391

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1340 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)(5,6,8,7)(9,10)(11,22)(12,18)(13,31)(14,21)(15,20)(16,25)(17,30)(19,32)(23,34)(24,36)(26,38)(27,35)(28,33)(29,37)(39,40);(5,7,8,6)(9,11,14,25,15,26,32,27)(10,16,19,22,20,35,21,38)(12,28,29,23,30,39,13,24)(17,36,37,40,18,34,31,33);(2,4)(9,12,14,29,15,30,32,13)(10,17,21,31,20,18,19,37)(11,23,27,28,26,24,25,39)(16,33,22,36,35,40,38,34);(9,13,32,30,15,29,14,12)(10,18,21,37,20,17,19,31)(11,24,27,39,26,23,25,28)(16,34,38,40,35,36,22,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275587);