#363703

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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 ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

656 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)(11,21,25,30)(12,26)(13,16)(14,19)(15,20)(17,24,33,23)(18,28)(22,36,37,32)(27,31)(29,34)(35,39,40,38);(2,5)(3,8)(10,16,20,19)(12,27)(17,30,33,21)(18,26,34,31)(22,37)(28,29)(32,35,36,40)(38,39);(1,3,4,8)(2,6,5,7)(9,11,13,23,15,25,14,24)(10,17,19,21,20,33,16,30)(12,22,28,38,27,37,29,39)(18,32,26,35,34,36,31,40);(2,5)(6,7)(9,12,13,28,15,27,14,29)(10,18,19,26,20,34,16,31)(11,22,23,38,25,37,24,39)(17,32,21,35,33,36,30,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270899);