#1291424

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fghi,d^2=f,e^2=h,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, 1198620);