#1311463

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6,10,12)(7,8,11,9)(14,17)(15,19)(21,22)(23,28)(24,29)(25,30)(26,27)(31,32)(33,35)(34,36);(1,2,3,4)(5,7,10,11)(6,8,12,9)(22,27)(23,30)(29,35)(31,36);(1,2)(3,4)(5,8,10,9)(6,7,12,11)(13,14,16,17)(15,18,19,20)(21,23,26,30)(22,28,27,25)(24,31,33,36)(29,32,35,34);(5,9,10,8)(6,11,12,7)(21,24)(22,29)(23,31)(25,34)(26,33)(27,35)(28,32)(30,36);(5,6)(7,8)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,28)(22,30,27,23)(24,32,33,34)(29,31,35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=i,e^2=f,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, 1218659);