#474251

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,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, 381447);