#383051

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2030 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)(5,6,7,8)(9,10,14,20,15,21,13,19)(11,22,26,16,27,34,25,35)(12,17,30,18,31,37,29,36)(23,38,28,32,24,33,40,39);(9,11,15,27)(10,16,21,35)(12,28,31,40)(13,26,14,25)(17,33,37,38)(18,32,36,39)(19,34,20,22)(23,29,24,30);(2,4)(9,12)(10,17)(11,23)(13,29)(14,30)(15,31)(16,32)(18,20)(19,36)(21,37)(22,38)(24,27)(25,40)(26,28)(33,34)(35,39);(2,4)(9,12)(10,18)(11,24)(13,30)(14,29)(15,31)(16,33)(17,20)(19,37)(21,36)(22,39)(23,27)(25,40)(26,28)(32,34)(35,38)

Polycyclic presentation:

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

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, 290247);