#4342860

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

1698 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)(5,6,10,9)(7,12,11,8)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,30)(31,38)(32,35)(33,41)(34,37)(36,42)(39,43)(40,44);(14,18)(16,25)(19,21)(22,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,7)(6,8)(9,12)(10,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32,34,31)(30,35,37,38)(33,39,42,40)(36,44,41,43);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44);(5,9,10,6)(7,12,11,8)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,43)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fh,d^2=fhi,e^2=g,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, 4250056);