#1465653

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

1738 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgi,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1372849);