#380992

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1172 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288188);