#583333

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1642 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)(5,6,8,10)(7,11,12,9)(15,21)(19,24)(22,27)(26,28)(29,30,34,35)(31,36,37,32)(33,41,42,38)(39,40,44,43);(5,6,8,10)(7,9,12,11)(14,18)(19,24)(20,25)(26,28)(30,35)(31,37)(38,41)(39,44);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,41)(42,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,36)(30,37,35,31)(33,40,42,43)(38,44,41,39);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,38)(31,39)(32,40)(34,42)(35,41)(36,43)(37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,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, 490529);