#4408412

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1710 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)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,32)(31,36,39,35)(33,40,41,37)(38,42,44,43);(15,21)(19,22)(23,27)(26,28)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(1,2)(3,4)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,32,34,30)(31,35,39,36)(33,37,41,40)(38,43,44,42);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,41)(30,37,32,40)(31,38,39,44)(35,42,36,43)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=h,d^2=fh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4315608);