#636365

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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

2234 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 543561);