#390802

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=efgh,d^2=e,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, 297998);