#1301631

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

1962 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)(5,6,8,10)(7,11,12,9)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(13,14,16,18)(15,17,19,20)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,35)(33,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,33)(22,28,25,32)(23,30,31,36)(27,34,29,35);(5,7,8,12)(6,9,10,11)(21,23)(22,29)(24,30)(25,27)(26,31)(28,35)(32,34)(33,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,17,18,20)(21,25,26,22)(23,29,31,27)(24,32,33,28)(30,35,36,34)

Polycyclic presentation:

2^9|a^2=i,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, 1208827);