#4499422

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4406618);