#1465609

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

1570 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=i,d^2=fh,e^2=fg,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, 1372805);