#5283249

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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

1690 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)(7,8)(9,12)(10,11)(13,14,18,17)(15,20,16,19)(21,22)(23,29)(24,32)(25,31)(26,27)(28,33)(30,34)(35,36);(5,6,9,12)(7,8,11,10)(14,17)(15,16)(22,27)(24,33)(25,29)(30,36);(7,11)(8,10)(13,15,18,16)(14,19,17,20)(21,23,26,31)(22,25,27,29)(24,30,33,36)(28,34,32,35);(1,2,3,4)(5,7,9,11)(6,10,12,8)(13,16,18,15)(14,19,17,20)(21,24)(22,28)(23,30)(25,34)(26,33)(27,32)(29,35)(31,36);(5,8,9,10)(6,11,12,7)(13,17,18,14)(15,19,16,20)(21,25)(22,23)(24,34)(26,29)(27,31)(28,30)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=g,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5190445);