#471346

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=egi,b^2=fh,c^2=ef,d^2=efi,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^c=de

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378542);