#582242

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2014 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)(14,18)(19,24)(20,25)(26,28)(30,34)(31,37)(32,40)(42,44);(6,9)(10,12)(14,18)(15,21)(20,25)(22,27)(29,30,33,34)(31,35,37,38)(32,36,40,41)(39,42,43,44);(5,6,8,9)(7,10,11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^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, 489438);