#390803

order	512
nilpotency class	3
isoclinism	isoclinic class of #219542
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

852 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)(3,6)(4,7)(5,8)(9,10,14,16,15,20,13,19)(11,21,24,17,25,31,22,32)(12,18,27,29,28,34,26,33)(23,35,37,30,38,39,36,40);(1,3,6,2)(4,8,5,7)(10,16)(11,22)(12,26)(13,14)(19,20)(21,31)(24,25)(27,28)(29,33)(30,39)(35,40)(36,37);(1,4,6,5)(2,7,3,8)(9,11)(10,17,20,32)(12,23)(13,24)(14,22)(15,25)(16,21,19,31)(18,30,34,40)(26,37)(27,36)(28,38)(29,35,33,39);(1,5,6,4)(2,7,3,8)(9,12,14,27,15,28,13,26)(10,18,16,29,20,34,19,33)(11,23,22,36,25,38,24,37)(17,30,21,35,32,40,31,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=efgh,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=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, 297999);