#1323273

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

1898 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)(13,14)(15,18)(16,20)(17,19)(21,22,26,23)(24,29,32,27)(25,28,34,30)(31,36,33,35);(2,4)(5,7,10,11)(6,8,12,9)(13,15,17,20)(14,18,19,16)(22,23)(27,29)(28,30)(35,36);(1,2)(3,4)(5,8,10,9)(6,7,12,11)(13,15)(14,16)(17,20)(18,19)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35);(5,9,10,8)(6,11,12,7)(21,24)(22,27)(23,29)(25,33)(26,32)(28,36)(30,35)(31,34);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=fi,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, 1230469);