#1281556

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,e^2=gh,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, 1188752);