#2747226

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2127 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2654422);