#473305

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 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)(4,8)(5,7)(9,10)(11,15)(12,16)(13,18)(14,26)(17,25)(19,20)(21,28)(22,30)(23,37)(24,32)(27,33)(29,35)(31,34)(36,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11,20,34,13,22,14,23)(10,15,26,37,18,30,19,31)(12,21,25,35,24,36,27,38)(16,28,33,39,32,40,17,29);(2,7)(6,8)(9,12,13,24)(10,16,18,32)(11,21)(14,25,20,27)(15,28)(17,19,33,26)(22,36)(23,35)(29,37)(30,40)(31,39)(34,38);(1,4,5,3)(2,8,7,6)(9,12)(10,17)(11,21)(13,24)(14,27)(15,29)(16,26)(18,33)(19,32)(20,25)(22,36)(23,38)(28,37)(30,39)(31,40)(34,35)

Polycyclic presentation:

2^9|b^2=efgh,c^2=e,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,g^d=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380501);