#466700

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

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

1012 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=f,d^2=f,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, 373896);