#4457338

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=fi,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4364534);