#393595

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 300791);