#583240

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

1662 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)(5,6,9,7)(8,11,12,10)(14,18)(15,21)(20,25)(22,27)(29,30,33,36)(31,34,39,37)(32,35,40,42)(38,41,44,43);(5,6)(7,9)(8,10)(11,12)(15,21)(19,24)(22,27)(26,28)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(1,3,4,2)(6,7)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,41)(40,42)(43,44);(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(5,8,9,12)(6,10,7,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=hi,d^2=h,e^2=g,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, 490436);