#4894727

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2320 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4801923);