#463426

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370622);