#1262558

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

1978 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=fh,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, 1169754);