#390801

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

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,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, 297997);