#2594780

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2392 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fghi,d^2=fi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2501976);