#467673

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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)(4,8)(5,7)(9,10)(11,21)(12,26,30,36)(13,19)(14,18)(15,20)(16,23)(17,29,35,28)(22,38,39,31)(24,33)(25,32)(27,34,40,37);(1,3,5,4)(2,6,7,8)(9,11,13,23,15,25,14,24)(10,16,18,21,20,33,19,32)(12,22,29,40,30,39,28,27)(17,31,36,37,35,38,26,34);(2,7)(6,8)(9,11,15,25)(10,16,20,33)(12,27,30,40)(13,24,14,23)(17,34,35,37)(18,32,19,21)(22,29,39,28)(26,31,36,38);(1,4,5,3)(2,8,7,6)(9,12,15,30)(10,17,20,35)(11,22,25,39)(13,29,14,28)(16,31,33,38)(18,36,19,26)(21,37,32,34)(23,40,24,27)

Polycyclic presentation:

2^9|a^2=eg,b^2=efg,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,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, 374869);