#156067

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2470 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)(5,6,8,7)(10,13)(11,15)(17,18)(19,21)(20,23)(22,24)(25,26,29,28)(27,31,32,30);(5,7,8,6)(10,13)(14,16)(18,19)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,22,24,23)(25,27,29,32)(26,30,28,31);(1,2)(3,4)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=efg,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, 63263);