#2046083

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1452 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)(3,8)(4,5)(6,7)(9,10,13,12)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(27,31)(29,32);(2,7)(5,8)(11,15)(14,16)(18,19)(20,21)(25,26,28,30)(27,29,31,32);(1,3,6,4)(2,8,7,5)(9,11,13,15)(10,14,12,16)(17,19,22,18)(20,24,21,23);(1,4,6,3)(2,5,7,8)(17,20,22,21)(18,23,19,24)(25,27)(26,29)(28,31)(30,32);(1,5)(2,3)(4,7)(6,8)(9,12,13,10)(11,16,15,14)(17,21,22,20)(18,24,19,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1953279);