#1226626

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

2338 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)(3,6,7,8)(9,10)(11,16)(12,14)(13,15)(18,21)(19,23)(25,26)(27,30)(28,32)(29,31);(2,4)(6,8)(9,11)(10,12)(13,14)(15,16)(25,27)(26,28)(29,32)(30,31);(3,7)(6,8)(11,14)(12,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,28,31,30);(1,3)(2,6)(4,8)(5,7)(9,12,13,16)(10,14,15,11)(17,18)(19,22)(20,21)(23,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,10)(11,16)(12,14)(13,15)(17,19,20,23)(18,22,21,24)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

2^9|a^2=fh,c^2=gi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

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, 1133822);