#470890

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

711 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)(9,10,15,28,13,17,14,19)(11,16,23,29,22,31,20,33)(12,24,27,32,25,18,26,34)(21,36,35,39,37,30,38,40);(1,3,5,4)(2,6,7,8)(9,11,13,22)(10,16)(12,21,25,37)(14,20,15,23)(17,31)(18,30)(19,33)(24,36)(26,38,27,35)(28,29)(32,39)(34,40);(1,4,5,3)(2,6,7,8)(10,17)(11,20,22,23)(12,25)(16,29,31,33)(19,28)(21,35,37,38)(26,27)(30,39,36,40);(9,12,13,25)(10,18,17,24)(11,21,22,37)(14,27,15,26)(16,30,31,36)(19,34,28,32)(20,35,23,38)(29,39,33,40)

Polycyclic presentation:

2^9|a^2=eg,b^2=fh,c^2=ef,d^2=i,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, 378086);