#732131

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

2148 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=i,e^2=g,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, 639327);