#451044

order	512
nilpotency class	3
isoclinism	isoclinic class of #417562
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1302 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)(11,21,25,16)(12,26)(13,18,22,38)(14,29)(15,19)(17,27)(20,32)(23,34)(24,35,30,39)(28,33,40,36)(31,37);(2,5)(3,7)(9,11)(10,16)(12,22)(13,23)(14,30)(15,25)(17,36)(18,26)(19,21)(20,39)(24,32)(27,28)(29,35)(31,40)(33,37)(34,38);(9,12,15,23)(10,17,19,37)(11,22,25,13)(14,31,32,27)(16,33,21,36)(18,35,38,39)(20,34,29,26)(24,28,30,40);(1,3,4,7)(2,6,5,8)(9,13,15,22)(10,18)(11,23,25,12)(14,28,32,40)(16,34)(17,35)(19,38)(20,36)(21,26)(24,31,30,27)(29,33)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358240);