#392313

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1340 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)(11,18)(12,17)(13,27,30,35)(14,21)(15,20)(16,22)(19,29,36,28)(23,37,39,31)(24,33)(25,32)(26,34,38,40);(2,5)(6,8)(9,11,15,24,16,25,14,12)(10,17,21,32,22,33,20,18)(13,23,29,38,30,39,28,26)(19,31,35,40,36,37,27,34);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,24,15,11)(10,18,20,33,22,32,21,17)(13,26,28,39,30,38,29,23)(19,34,27,37,36,40,35,31);(9,13)(10,19)(11,23)(12,26)(14,28)(15,29)(16,30)(17,31)(18,34)(20,27)(21,35)(22,36)(24,38)(25,39)(32,40)(33,37)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^d=gi,h^d=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, 299509);