#387108

order	512
nilpotency class	3
isoclinism	isoclinic class of #213760
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, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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:

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

Polycyclic presentation:

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

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, 294304);