#732106

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2070 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:

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

Polycyclic presentation:

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

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, 639302);