#272563

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

776 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)(3,12)(4,8)(5,13)(6,10)(7,11)(9,16)(14,15)(17,18,22,26,23,27,21,35)(19,28,30,24,31,39,36,40)(20,32,33,42,34,41,37,25)(29,44,46,48,47,38,45,43);(1,3,7,14)(2,5,11,4)(6,8,16,13)(9,15,10,12)(17,19,23,31)(18,24,27,40)(20,29,34,47)(21,36,22,30)(25,38,42,44)(26,39,35,28)(32,43,41,48)(33,46,37,45);(1,4)(2,8)(3,9)(5,7)(6,15)(10,14)(11,13)(12,16)(17,20,23,34)(18,25,27,42)(19,29,31,47)(21,33,22,37)(24,38,40,44)(26,41,35,32)(28,43,39,48)(30,45,36,46);(1,5)(2,3)(4,7)(6,15)(8,9)(10,13)(11,14)(12,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,39)(27,40)(28,35)(32,43)(33,46)(34,47)(37,45)(41,48)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179759);