#5956459

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2210 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863655);