#4462423

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1758 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)(13,14,18,19)(15,20,21,16)(17,25,26,22)(23,24,28,27)(29,30,35,33)(31,36,32,39)(34,37,42,41)(38,43,40,44);(2,4)(14,19)(15,21)(22,25)(23,28)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(2,4)(13,15)(14,20)(16,19)(17,23)(18,21)(22,27)(24,25)(26,28)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,20)(14,21,19,15)(17,24,26,27)(22,28,25,23)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,22,19,25)(15,23,21,28)(16,24,20,27)(29,34,35,42)(30,37,33,41)(31,38,32,40)(36,43,39,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=fgh,d^2=fg,e^2=fghi,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, 4369619);