#369299

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=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, 276495);