#427998

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335194);