#1316561

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1826 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,14)(13,15)(18,21)(19,23)(25,26,29,32)(27,31,28,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(18,21)(22,24);(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(25,27,29,28)(26,30,32,31);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=i,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, 1223757);