#4309091

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
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

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(6,9)(10,12)(15,21)(16,23)(18,24)(19,26)(29,30,35,32)(31,38,36,33)(34,40,42,37)(39,41,43,44);(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,31,35,36)(30,33,32,38)(34,39,42,43)(37,41,40,44);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,32,35,30)(31,33,36,38)(34,40,42,37)(39,41,43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,18,20,24)(16,22,23,27)(19,25,26,28)(29,33)(30,36)(31,32)(34,41)(35,38)(37,43)(39,40)(42,44);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

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, 4216287);