#5026756

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3584 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)(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,19)(20,27)(24,26)(28,29);(9,11)(10,13)(12,16)(14,15)(17,18,22,19)(20,23,27,25)(21,24,30,26)(28,31,29,32);(2,3)(10,12)(11,15)(17,19,22,18)(20,23,27,25)(21,24,30,26)(28,32,29,31);(1,3,4,2)(5,6)(7,8)(9,12,14,10)(11,16,15,13)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31);(9,13,14,16)(10,15,12,11)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=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, 4933952);