#289433

order	512
nilpotency class	3
isoclinism	isoclinic class of #289395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=efh,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,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196629);