#593156

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 500352);