#4468318

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 2097152
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

1886 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)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30)(31,32)(33,41)(34,35)(36,42)(37,40)(38,44)(39,43);(5,6,8,10)(7,9,12,11)(15,21)(19,22)(23,27)(26,28)(30,35)(31,37)(33,42)(39,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31,34,37)(30,32,35,40)(33,38,42,43)(36,39,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,31)(33,39)(34,40)(35,37)(36,38)(41,43)(42,44);(5,7)(6,9)(8,12)(10,11)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,36)(31,38)(32,39)(34,42)(35,41)(37,43)(40,44)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=f,d^2=fgh,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, 4375514);