#155668

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2898 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62864);