#5037177

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1664 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)(5,6,9,7)(8,10,12,11)(13,14,18,19)(15,20,17,16)(21,22,27,26)(23,29,24,28)(25,33,34,30)(31,35,32,36);(8,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,23,27,24)(22,28,26,29)(25,32,34,31)(30,36,33,35);(6,7)(10,11)(14,19)(15,17)(21,24,27,23)(22,29,26,28)(25,32,34,31)(30,36,33,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,15,19,17)(21,25,27,34)(22,30,26,33)(23,31,24,32)(28,35,29,36);(5,8)(6,10)(7,11)(9,12)(13,17,18,15)(14,20,19,16)(21,26,27,22)(23,28,24,29)(25,33,34,30)(31,35,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4944373);