#363214

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2468 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,f^c=fi,h^d=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, 270410);