#2606856

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2557 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)(10,11)(14,15)(17,18)(19,22)(20,23)(21,26)(24,30)(25,27)(28,32)(29,31);(6,8)(9,10,13,11)(12,14,16,15)(18,19)(20,27)(24,26)(28,29);(5,6)(7,8)(17,19,22,18)(20,25,27,23)(21,26,30,24)(28,32,29,31);(1,2)(3,4)(9,11,13,10)(12,14,16,15)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31);(9,12,13,16)(10,14,11,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=g,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, 2514052);