#466732

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1404 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)(3,4)(6,7)(9,10)(11,21)(12,26)(13,19)(14,17)(15,20)(16,24)(18,27)(22,31)(23,37)(25,32)(28,33)(29,34)(30,36)(35,39)(38,40);(9,11)(10,16,20,32)(12,23)(13,24)(14,25)(15,22)(17,21,19,31)(18,30,33,37)(26,35,34,40)(27,39)(28,38)(29,36);(5,6,8,7)(10,17)(11,22)(12,27)(13,14)(16,21)(18,33)(19,20)(23,38)(28,29)(30,37)(31,32)(36,39);(2,4)(5,7,8,6)(9,12,14,28,15,29,13,27)(10,18,19,34,20,33,17,26)(11,23,25,38,22,36,24,39)(16,30,31,40,32,37,21,35)

Polycyclic presentation:

2^9|b^2=gi,c^2=ef,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,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, 373928);