#1301650

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

1914 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=f,d^2=i,e^2=fh,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, 1208846);