#582256

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fh,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, 489452);