#364169

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1440 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)(11,19)(12,25,29,36)(13,30)(14,31)(15,20)(16,21)(17,23)(18,28,37,32)(22,33,26,34)(24,38)(27,35,39,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,13,16,24,14,23)(10,17,31,38,21,30,20,19)(12,26,32,39,29,22,28,27)(18,34,25,35,37,33,36,40);(3,8)(5,7)(9,12)(10,18)(11,22)(13,27)(14,28)(15,32)(16,29)(17,33)(19,35)(20,36)(21,37)(23,39)(24,26)(25,31)(30,34)(38,40);(2,6)(5,7)(9,13,15,24,16,23,14,11)(10,19,31,17,21,38,20,30)(12,27,32,26,29,39,28,22)(18,35,25,33,37,40,36,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,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,f^c=fi,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, 271365);