#1226621

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

2402 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,20)(19,21,24,23)(25,26,29,28)(27,31,32,30);(2,5)(6,7)(10,14)(12,15)(17,19)(18,23)(20,21)(22,24)(25,27)(26,30)(28,31)(29,32);(3,8)(6,7)(10,14)(11,16)(18,20)(19,24)(25,27,29,32)(26,31,28,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,20,22,18)(19,21,24,23)(25,27)(26,31)(28,30)(29,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,15)(10,11,14,16)(17,21)(18,19)(20,24)(22,23)(25,28,29,26)(27,30,32,31)

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=fghi,e^2=gh,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, 1133817);