#509614

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=ehi,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,f^d=fi

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, 416810);