#1300072

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1650 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=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, 1207268);