#371758

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
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	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278954);