#2606567

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2356 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)(9,10)(11,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(6,8)(10,15)(11,12)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(5,6)(7,8)(9,11,14,12)(10,16,15,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,2)(3,4)(9,12,14,11)(10,16,15,13)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(9,13,14,16)(10,12,15,11)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=f,e^2=f,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, 2513763);