#454432

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1864 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)(9,10,15,20)(11,21,26,36)(12,27,29,17)(13,18,32,38)(14,33,22,19)(16,30,34,25)(23,35,31,40)(24,39,28,37);(5,6)(7,8)(9,11,22,30,15,26,14,25)(10,16,33,36,20,34,19,21)(12,13,24,23,29,32,28,31)(17,35,39,18,27,40,37,38);(2,4)(6,8)(9,12)(10,17)(11,23)(13,30)(14,24)(15,29)(16,18)(19,39)(20,27)(21,40)(22,28)(25,32)(26,31)(33,37)(34,38)(35,36);(9,13,14,31,15,32,22,23)(10,18,19,35,20,38,33,40)(11,24,25,12,26,28,30,29)(16,27,21,39,34,17,36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 361628);