#4464168

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

1920 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)(5,6,8,10)(7,11,12,9)(15,21)(16,23)(19,24)(20,25)(29,30)(31,35)(32,34)(33,40)(36,41)(37,39)(38,44)(42,43);(5,6)(7,9)(8,10)(11,12)(14,18)(19,24)(20,25)(26,28)(30,32)(31,37)(36,40)(38,43);(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,37)(30,35,32,39)(33,38,41,43)(36,42,40,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,35,37,39)(33,40,41,36)(38,42,43,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,41)(30,36,32,40)(31,38,37,43)(35,42,39,44)

Polycyclic presentation:

2^9|a^2=g,c^2=fg,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,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, 4371364);