#390642

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

807 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=gh,d^2=e,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, 297838);