#273273

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
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	#121	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13,16,10)(4,9)(5,20,21,8)(6,11)(7,12)(14,24,31,29)(15,28)(17,23,30,32)(18,25)(19,26)(22,27)(33,34)(35,44)(36,50)(37,53)(38,42)(39,43)(40,47)(41,54)(45,49)(46,61)(48,56)(51,58)(52,57)(55,63)(59,64)(60,62);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,23)(10,11)(12,20)(13,28)(17,18)(19,31)(22,30)(24,27)(25,29)(26,32)(33,35,39,49)(34,40,43,56)(36,46,52,63)(37,48,38,47)(41,55,58,61)(42,45,53,44)(50,60,57,59)(51,64,54,62);(1,4,7,19)(2,9,12,26)(3,14,16,31)(5,17,21,30)(6,22,15,18)(8,23,20,32)(10,24,13,29)(11,27,28,25)(33,34)(35,45)(36,41)(37,42)(38,53)(39,43)(40,47)(44,49)(46,60)(48,56)(50,54)(51,57)(52,58)(55,64)(59,63)(61,62);(1,5,7,21)(2,10,12,13)(3,15,16,6)(4,17,19,30)(8,11,20,28)(9,24,26,29)(14,18,31,22)(23,27,32,25)(33,36,39,52)(34,41,43,58)(35,46,49,63)(37,54,38,51)(40,55,56,61)(42,50,53,57)(44,59,45,60)(47,64,48,62)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=i,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180469);