#582203

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1826 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)(7,11)(10,12)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,26,28,27)(30,31)(32,39)(33,41)(42,43);(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(29,30,34,31)(32,37,39,35)(33,36,41,38)(40,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,42,44,43);(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(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=g,c^2=f,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, 489399);