#387067

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
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

1013 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=eg,d^2=i,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,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, 294263);