#466295

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

864 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 373491);