#368394

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

1330 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)(12,19)(13,29)(14,30)(15,21)(16,20)(17,24)(18,28)(23,36)(25,35)(26,38)(27,37)(31,34)(32,40)(33,39);(5,7,8,6)(9,11,14,24,15,25,16,31)(10,17,20,22,21,34,30,35)(12,26,27,39,28,40,13,23)(18,36,37,32,19,33,29,38);(2,4)(9,12,16,13,15,28,14,27)(10,18,20,37,21,19,30,29)(11,23,24,26,25,39,31,40)(17,32,35,36,34,38,22,33);(9,13,16,28,15,27,14,12)(10,19,30,37,21,18,20,29)(11,23,31,40,25,39,24,26)(17,33,35,32,34,36,22,38)

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,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, 275590);