#4462420

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

1690 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)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(1,2,3,4)(14,15)(19,21)(20,22)(26,27)(30,34)(31,38)(36,40)(39,44);(1,2,3,4)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=gh,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, 4369616);