#1271386

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2795 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=fgh,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, 1178582);