#471363

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

830 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18,20,22)(19,21,23,24)(25,26,30,32)(27,29,31,28);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,19)(18,21)(20,23)(22,24)(25,27,30,31)(26,28,32,29);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,16,14)(19,23)(21,24)(25,28,27,32,30,29,31,26);(3,8)(6,7)(9,12,13,11)(10,15,16,14)(25,29,31,32,30,28,27,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=efh,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^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, 378559);