#586773

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

1626 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)(6,7)(10,11)(14,18)(16,23)(19,24)(22,27)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(5,6,9,7)(8,11,12,10)(15,21)(19,24)(22,27)(26,28)(30,32)(33,41)(35,37)(38,44);(6,7)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(5,8)(6,10)(7,11)(9,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,d^2=f,e^2=fgh,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, 493969);