#1337724

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2340 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=f,e^2=hi,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, 1244920);