#187553

order	512
nilpotency class	3
isoclinism	isoclinic class of #187428
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	#62762	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2022 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)(3,12,16,8)(4,17,19,24)(5,20)(6,10)(7,11)(9,18,25,21)(13,28)(14,23,15,27)(22,29)(26,31)(30,32)(33,34,38,44,39,42,37,43)(35,45,49,40,48,55,47,54)(36,41,50,57,51,56,52,58)(46,60,61,64,62,59,63,53);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,22)(10,23)(11,12)(17,32)(18,29)(19,30)(20,27)(21,31)(24,28)(25,26)(33,35)(34,40)(36,46)(37,49)(38,47)(39,48)(41,53)(42,54)(43,55)(44,45)(50,63)(51,62)(52,61)(56,64)(57,59)(58,60);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,29)(15,31)(16,30)(17,20)(23,28)(27,32)(33,36)(34,41)(35,46)(37,52)(38,50)(39,51)(40,53)(42,56)(43,58)(44,57)(45,59)(47,63)(48,62)(49,61)(54,64)(55,60);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,23,12,27)(9,24,25,17)(13,29,30,31)(22,28,26,32)(34,42)(40,54)(41,56)(43,44)(45,55)(53,64)(57,58)(59,60)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,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, 94749);