#5021454

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
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

1842 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4928650);