#1335715

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

2732 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)(6,8)(9,10)(11,16)(12,14)(13,15)(18,21)(19,23)(25,26)(27,32)(28,30)(29,31);(9,11,13,12)(10,14,15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,30)(26,28,31,32);(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,21)(22,23)(25,26)(27,28)(29,31)(30,32);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|b^2=fg,e^2=fhi,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.

Relationships

maximal subgroup of	#72336336, #72336368, #72336449
maximal normal subgroup of	#72336336, #72336368, #72336449

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1242911);