#4416901

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
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

1770 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)(5,6,8,10)(7,11,12,9)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30,35,32)(31,36,33,39)(34,40,42,37)(38,44,41,43);(14,15)(19,21)(20,22)(26,27)(29,31,35,33)(30,36,32,39)(34,38,42,41)(37,43,40,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,32,35,30)(31,36,33,39)(34,40,42,37)(38,43,41,44);(5,7,8,12)(6,9,10,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,33,35,31)(30,36,32,39)(34,41,42,38)(37,43,40,44);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,34)(30,37)(31,38)(32,40)(33,41)(35,42)(36,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4324097);