#3143259

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
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)(6,9)(7,11)(13,14,17,16)(15,18,19,20)(21,22)(23,30)(24,26)(25,33)(27,29)(28,35)(31,36)(32,34);(6,9)(10,12)(15,19)(18,20)(21,23,27,24)(22,26,29,30)(25,31,35,32)(28,36,33,34);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20)(21,24,27,23)(22,26,29,30)(25,31,35,32)(28,34,33,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,18,19,20)(21,25)(22,28)(23,31)(24,32)(26,36)(27,35)(29,33)(30,34);(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,26)(22,23)(24,29)(25,34)(27,30)(28,32)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fgh,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3050455);