#512375

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=gh,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, 419571);