#363102

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 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)(3,7)(4,5)(6,8)(9,10)(11,20,23,16)(12,24,26,38)(13,18)(14,19)(15,22,29,37)(17,25,32,39)(21,34)(27,31)(28,36)(30,35)(33,40);(2,5)(3,8)(9,11)(10,15,19,29)(12,21)(13,22)(14,23)(16,18,20,31)(17,28,32,34)(24,33,38,30)(25,35)(26,36)(27,37)(39,40);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,18)(17,30)(19,20)(24,28)(25,35)(26,36)(27,37)(29,31)(32,33)(34,38)(39,40);(2,5)(6,7)(9,12,13,25,14,26,27,39)(10,17,31,38,19,32,18,24)(11,21,22,35,23,36,37,40)(15,28,16,30,29,34,20,33)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,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, 270298);