#326542

order	512
nilpotency class	3
isoclinism	isoclinic class of #326535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,16,5,10,6,11)(3,4,13,15,9,8,12,14)(17,18,22,28)(19,25)(20,32,34,38)(21,24,23,27)(26,35,42,33)(29,37)(30,44)(31,39)(36,41)(40,43)(45,47)(46,48);(1,3,5,9)(2,8,10,4)(6,12,7,13)(11,15,16,14)(18,24,28,27)(19,29,31,36)(25,39)(26,38,42,32)(30,43,45,46)(37,41)(40,48)(44,47);(1,4,5,8)(2,9,10,3)(6,15,7,14)(11,12,16,13)(17,19)(18,25)(20,30)(21,29)(22,31)(23,36)(24,37)(26,40)(27,41)(28,39)(32,44)(33,43)(34,45)(35,46)(38,47)(42,48);(2,10)(3,9)(11,16)(12,13)(17,20,23,35,22,34,21,33)(18,26,24,38,28,42,27,32)(19,30,36,46,31,45,29,43)(25,40,37,47,39,48,41,44)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=di,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233738);