#454285

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1398 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)(3,4)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,29,30,26)(20,32,31,25)(22,28,23,27);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(18,24)(25,32)(26,29)(27,28);(2,4)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,30)(18,25,24,32)(20,23,31,22)(26,28,29,27);(10,14)(13,15)(17,20,21,31)(18,26,24,29)(19,22,30,23)(25,27,32,28)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,d^2=i,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,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361481);