#582261

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=f,e^2=fg,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, 489457);