#3829716

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1826 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fhi,d^2=gi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3736912);