#1296469

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1203665);