#763796

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

2044 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)(7,11)(8,12)(13,14,17,20)(15,18,16,19)(21,22,25,26)(23,27,30,34)(24,31,33,28)(29,36,32,35);(5,6,10,9)(7,8,11,12)(13,14,17,20)(15,18,16,19)(22,26)(27,34)(28,31)(35,36);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,19)(17,20)(21,23,25,30)(22,27,26,34)(24,32,33,29)(28,36,31,35);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(13,15,17,16)(14,18,20,19)(21,22)(23,27)(24,28)(25,26)(29,35)(30,34)(31,33)(32,36);(5,9,10,6)(7,12,11,8)(13,16,17,15)(14,19,20,18)(21,24,25,33)(22,28,26,31)(23,29,30,32)(27,35,34,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 670992);