#467603

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

705 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)(4,8)(5,7)(9,10,15,20)(11,17,12,24)(13,27)(14,32,21,19)(16,26,34,23)(18,30)(22,35)(25,39)(28,33)(29,36)(31,37)(38,40);(1,3,5,4)(2,6,7,8)(9,11,21,26,15,12,14,23)(10,16,32,24,20,34,19,17)(13,22,29,38,30,25,31,28)(18,33,36,39,27,40,37,35);(2,7)(6,8)(9,12,15,11)(10,17,20,24)(13,28,30,38)(14,23,21,26)(16,32,34,19)(18,33,27,40)(22,29,25,31)(35,37,39,36);(1,4,5,3)(2,8,7,6)(9,13,15,30)(10,18,20,27)(11,22,12,25)(14,31,21,29)(16,33,34,40)(17,35,24,39)(19,37,32,36)(23,28,26,38)

Polycyclic presentation:

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

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, 374799);