#393811

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,h^b=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301007);