#4433069

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1778 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)(5,6,9,7)(8,11,12,10)(16,23)(19,26)(21,27)(24,28)(29,30,33,36)(31,37,39,34)(32,35,41,44)(38,43,40,42);(13,14,17,20)(15,18,22,25)(16,19,23,26)(21,24,27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,43)(41,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,43)(41,44);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15,17,22)(14,18,20,25)(16,21,23,27)(19,24,26,28)(29,31,33,39)(30,34,36,37)(32,40,41,38)(35,43,44,42);(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=fgi,b^2=h,c^2=gi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4340265);