#471113

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

549 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)(9,10,15,27,13,18,14,19)(11,16,21,31,22,30,26,35)(12,17,28,38,24,33,25,34)(20,29,23,32,36,39,37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16,18,30)(12,23)(13,22)(14,26)(15,21)(17,32,33,40)(19,35,27,31)(20,25)(24,37)(28,36)(29,38,39,34);(2,7)(6,8)(9,12)(10,17)(11,20,22,36)(13,24)(14,25)(15,28)(16,29,30,39)(18,33)(19,34)(21,23,26,37)(27,38)(31,32,35,40);(1,4,5,3)(2,8,7,6)(10,18)(12,24)(14,15)(16,30)(21,26)(23,37)(29,39)(34,38)

Polycyclic presentation:

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

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, 378309);