#264685

order	512
nilpotency class	3
isoclinism	isoclinic class of #264659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18222	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

843 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)(3,10)(4,8)(5,7)(6,12)(9,14)(11,16)(13,15)(17,18)(19,27)(20,31)(21,35)(22,26)(23,29)(24,34)(25,33)(28,41)(30,39)(32,45)(36,43)(37,42)(38,46)(40,44)(47,48);(1,3,4,12)(2,6,8,10)(5,11,14,13)(7,15,9,16)(17,19,21,29,22,30,33,43)(18,23,25,27,26,36,35,39)(20,28,34,44,32,42,46,48)(24,37,31,40,38,41,45,47);(2,7)(3,11)(5,14)(8,9)(12,13)(15,16)(20,32)(23,36)(24,38)(27,39)(28,42)(31,45)(34,46)(44,48);(2,8)(6,10)(7,9)(15,16)(17,20,33,46,22,32,21,34)(18,24,35,45,26,38,25,31)(19,28,43,48,30,42,29,44)(23,37,39,47,36,41,27,40)

Polycyclic presentation:

2^9|b^2=e,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gi,d^c=dh,f^c=fi,g^c=gi

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, 171881);