#732139

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

2246 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)(21,22,25,29)(23,27,31,26)(24,28,33,36)(30,35,32,34);(2,4)(14,17)(18,20)(21,23,25,31)(22,26,29,27)(24,32,33,30)(28,35,36,34);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,33)(34,36);(5,7,9,6)(8,11,12,10)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,35)(33,36);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=fi,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, 639335);