#119004

order	512
nilpotency class	3
isoclinism	isoclinic class of #119003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26200);