#606836

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1722 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)(5,6,8,10)(7,11,12,9)(14,16)(18,20)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36);(5,6,8,10)(7,9,12,11)(15,19)(18,20)(22,24)(25,33)(27,29)(30,36);(5,6)(7,9)(8,10)(11,12)(13,14,17,16)(15,18,19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,7,8,12)(6,9,10,11)(13,16,17,14)(15,20,19,18)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=f,e^2=fg,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, 514032);