#119141

order	512
nilpotency class	3
isoclinism	isoclinic class of #119140
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

503 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)(35,44,47,56)(36,48)(37,42)(38,53)(39,43)(40,46,57,52)(41,50)(45,58,49,55)(51,59)(54,60)(61,62,64,63);(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)(34,40)(36,49)(37,52)(38,46)(39,47)(41,58)(42,56)(43,57)(44,53)(45,51)(48,64)(50,62)(54,63)(55,60)(59,61);(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)(33,36,37,50,39,51,38,54)(34,41,42,59,43,60,53,48)(35,45,46,62,47,49,52,63)(40,55,44,61,57,58,56,64)

Polycyclic presentation:

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

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, 26337);