#705458

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

2046 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=hi,d^2=fg,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, 612654);