#587431

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

1810 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)(7,11)(10,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(1,3,4,2)(6,9)(10,12)(15,21)(18,25)(22,27)(26,28)(30,32)(33,41)(35,37)(38,44);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=f,d^2=f,e^2=g,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, 494627);