#464236

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1723 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)(6,7)(9,10)(11,17)(12,27)(13,29)(14,20)(15,19)(16,21)(18,31)(22,23)(24,36)(25,35)(26,37)(28,39)(30,32)(33,34)(38,40);(5,6,8,7)(9,11,15,25,16,26,14,24)(10,17,19,35,21,37,20,36)(12,28,30,38,18,34,29,22)(13,23,27,39,32,40,31,33);(2,4)(6,7)(9,12,16,18)(10,13,21,32)(11,22,26,38)(14,30,15,29)(17,33,37,39)(19,31,20,27)(23,35,40,36)(24,28,25,34);(5,7,8,6)(9,13)(10,18)(11,23)(12,21)(14,31)(15,27)(16,32)(17,34)(19,29)(20,30)(22,35)(24,33)(25,39)(26,40)(28,37)(36,38)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 371432);