#187566

order	512
nilpotency class	3
isoclinism	isoclinic class of #187442
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	#62831	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1997 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)(3,13,17,8)(4,18,21,25)(5,10)(6,12)(7,11)(9,20,26,19)(14,28)(15,24,16,23)(22,30)(27,29)(31,32)(33,34,38,43,39,42,37,44)(35,45,47,40,48,56,53,55)(36,49,52,41,50,57,51,58)(46,60,62,64,61,54,63,59);(1,3,7,17)(2,8,11,13)(4,14,21,31)(5,15,6,16)(9,22,26,27)(10,23,12,24)(18,32,25,28)(19,29,20,30)(33,35,39,48)(34,40,42,55)(36,46,50,61)(37,53,38,47)(41,54,58,60)(43,56,44,45)(49,64,57,59)(51,63,52,62);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,18)(11,26)(12,25)(13,27)(15,30)(16,29)(17,31)(23,32)(24,28)(33,36)(34,41)(35,46)(37,52)(38,51)(39,50)(40,54)(42,58)(43,49)(44,57)(45,59)(47,63)(48,61)(53,62)(55,60)(56,64);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,21,20)(8,23,13,24)(9,25,26,18)(14,29,31,30)(22,28,27,32)(34,42)(36,50)(40,55)(43,44)(45,56)(46,61)(51,52)(62,63)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,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, 94762);