#606818

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1882 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

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, 514014);