#454426

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	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1912 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)(11,23)(12,29)(13,19)(14,22)(15,21)(16,20)(17,26)(18,32)(24,39)(25,38)(27,35)(28,34)(30,40)(31,37)(33,36);(5,6)(7,8)(9,11,14,26,15,27,16,28)(10,17,20,23,21,34,22,35)(12,30,31,24,32,13,25,33)(18,36,37,19,29,39,38,40);(2,4)(6,8)(9,12)(10,18)(11,24)(13,28)(14,25)(15,32)(16,31)(17,19)(20,38)(21,29)(22,37)(23,36)(26,30)(27,33)(34,40)(35,39);(9,13,14,33,15,30,16,24)(10,19,20,39,21,40,22,36)(11,25,26,12,27,31,28,32)(17,29,23,38,34,18,35,37)

Polycyclic presentation:

2^9|b^2=e,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, 361622);