#4894737

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

2222 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,d^2=fgi,e^2=f,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, 4801933);