#4378361

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1762 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)(6,9)(10,12)(13,14)(15,21)(16,24)(17,18)(19,22)(20,26)(23,27)(25,28)(30,31)(32,39)(36,38)(40,44);(14,17)(15,22)(20,24)(23,25)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,43,44,42);(1,2)(3,4)(13,15)(14,19)(16,25)(17,21)(18,22)(20,28)(23,26)(24,27)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,42,44,43);(5,6,8,9)(7,10,11,12)(13,16)(14,20)(15,23)(17,24)(18,26)(19,27)(21,28)(22,25)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,21,22,19)(16,24,26,20)(23,28,25,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fh,d^2=hi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

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, 4285557);