#451272

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358468);