#5083043

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

1672 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,16,20,18)(21,22,26,29)(23,30,28,25)(24,32,35,27)(31,34,33,36);(7,11)(9,12)(13,15,17,20)(14,18,19,16)(23,28)(25,30)(31,33)(34,36);(5,6,8,10)(7,9,11,12)(13,14,17,19)(15,16,20,18)(21,23,26,28)(22,25,29,30)(24,33,35,31)(27,36,32,34);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,24,26,35)(22,27,29,32)(23,31,28,33)(25,34,30,36);(5,7,8,11)(6,9,10,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,28)(23,29)(24,34)(26,30)(27,33)(31,32)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=fi,e^2=fgh,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, 4990239);