#5021470

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

1810 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)(7,8)(9,10)(11,12)(15,20)(16,18)(21,22)(23,27)(24,28)(25,26)(29,34)(30,33)(31,32)(35,36);(13,14,17,19)(15,18,20,16)(22,26)(27,33)(28,31)(34,35);(6,10)(7,12)(13,15,17,20)(14,16,19,18)(21,22)(23,27)(24,31)(25,26)(28,32)(29,35)(30,33)(34,36);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,23,25,30)(22,27,26,33)(24,29,32,36)(28,34,31,35);(5,8,9,11)(6,7,10,12)(13,16,17,18)(14,15,19,20)(21,24,25,32)(22,28,26,31)(23,29,30,36)(27,34,33,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=fh,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, 4928666);