#364364

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2280 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 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)(4,6)(5,8)(9,10)(11,22)(12,26,29,34)(13,30)(14,20)(15,21)(16,25)(17,28,35,37)(18,31)(19,23)(24,32,27,33)(36,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,11,23,31,15,13,14,25)(10,16,20,30,21,18,19,22)(12,27,37,38,29,24,28,39)(17,33,26,36,35,32,34,40);(2,6)(3,8)(9,12,15,29)(10,17)(11,24,13,27)(14,28,23,37)(16,32)(18,33)(19,34)(20,26)(21,35)(22,36)(25,38,31,39)(30,40);(2,6)(5,7)(9,13,23,25,15,11,14,31)(10,18,20,22,21,16,19,30)(12,27,37,38,29,24,28,39)(17,33,26,36,35,32,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271560);