#1368912

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

1556 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fg,d^2=fgh,e^2=fhi,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, 1276108);