#363272

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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2466 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:

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

Polycyclic presentation:

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

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, 270468);