#363257

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2423 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:

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

Polycyclic presentation:

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

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, 270453);