#476504

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1321 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)(5,6)(7,8)(9,10,14,19,16,21,15,20)(11,22,25,37,27,36,26,17)(12,18,29,34,30,13,31,33)(23,39,35,38,28,24,40,32);(6,7)(9,11,16,27)(10,17,21,37)(12,28,30,23)(13,32,18,38)(14,26,15,25)(19,36,20,22)(24,34,39,33)(29,35,31,40);(5,7,8,6)(9,12,14,29,16,30,15,31)(10,18,19,34,21,13,20,33)(11,23,25,35,27,28,26,40)(17,32,22,39,37,38,36,24);(2,4)(6,7)(9,13,16,18)(10,12,21,30)(11,24,27,39)(14,34,15,33)(17,35,37,40)(19,31,20,29)(22,23,36,28)(25,38,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383700);