#2606859

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2456	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3467 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2514055);