#187860

order	512
nilpotency class	3
isoclinism	isoclinic class of #187780
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3565	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1996 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:

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

Polycyclic presentation:

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

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, 95056);