#456668

order	512
nilpotency class	3
isoclinism	isoclinic class of #456668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16284
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1405 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363864);