#4342097

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
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

1740 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4249293);