#456934

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1208 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364130);