#348516

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
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

750 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:

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

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,d^b=di,f^c=fi,g^c=gh,f^d=fh

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, 255712);