#1043784

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=i,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 950980);