#187867

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3553	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2020 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)(3,11,14,7)(4,15,16,26)(5,10)(6,9)(8,17,23,19)(12,28)(13,25,18,22)(20,24)(21,30)(27,31)(29,32)(33,34)(35,43,46,56)(36,47,49,40)(37,41)(38,42)(39,45,54,50)(44,59)(48,57,51,55)(52,58)(53,60)(61,64)(62,63);(1,3,6,14)(2,7,9,11)(4,12,16,29)(5,18,20,13)(8,21,23,27)(10,25,24,22)(15,28,26,32)(17,31,19,30)(33,35)(34,39)(36,44)(37,50)(38,46)(40,53)(41,56)(42,54)(43,58)(45,52)(47,63)(48,62)(49,61)(51,60)(55,59)(57,64);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,23)(10,26)(11,27)(13,31)(14,29)(15,24)(17,20)(18,30)(22,28)(25,32)(33,36)(34,40)(35,44)(37,51)(38,49)(39,53)(41,57)(42,47)(43,59)(45,62)(46,61)(48,52)(50,60)(54,63)(55,58)(56,64);(2,9)(7,11)(8,23)(10,24)(15,26)(21,27)(22,25)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95063);