#347906

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,f^c=fi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255102);