#364810

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 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)(4,5)(6,8)(9,10)(11,20)(12,24,26,36)(13,18)(14,19)(15,30)(16,22)(17,27,35,28)(21,34,38,40)(23,33)(25,31,39,37)(29,32);(2,5)(6,7)(9,11)(10,16)(12,25,26,39)(13,22)(14,29)(15,23)(17,34,35,40)(18,20)(19,33)(21,28,38,27)(24,37,36,31)(30,32);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,31)(18,35)(19,36)(20,37)(22,38)(23,39)(24,30)(25,29)(32,34)(33,40);(1,3,4,8)(2,6,5,7)(10,18)(16,20)(17,35)(19,30)(24,36)(31,37)(32,33)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,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, 272006);