#363271

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

2396 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,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, 270467);