#5035105

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1778 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fgi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=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, 4942301);