#187528

order	512
nilpotency class	3
isoclinism	isoclinic class of #187410
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	#62838	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2003 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)(4,15,16,24)(5,10)(6,9)(8,17,22,18)(12,27)(13,26,31,21)(19,23)(20,29)(25,30)(28,32)(33,34,38,43,39,44,37,42)(35,45,47,40,49,54,48,55)(36,41,51,57,52,58,50,56)(46,60,61,64,62,59,63,53);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,32)(16,28)(17,29)(18,30)(19,31)(22,25)(23,26)(24,27)(33,35)(34,40)(36,46)(37,47)(38,48)(39,49)(41,53)(42,54)(43,45)(44,55)(50,61)(51,63)(52,62)(56,60)(57,59)(58,64);(1,4,6,16)(2,8,9,22)(3,12,14,28)(5,18,19,17)(7,20,11,25)(10,24,23,15)(13,30,31,29)(21,27,26,32)(33,36)(34,41)(35,46)(37,50)(38,51)(39,52)(40,53)(42,56)(43,57)(44,58)(45,59)(47,61)(48,63)(49,62)(54,60)(55,64);(4,16)(8,22)(12,28)(15,24)(17,18)(20,25)(27,32)(29,30)(33,37,39,38)(34,42,44,43)(35,47,49,48)(36,50,52,51)(40,54,55,45)(41,56,58,57)(46,61,62,63)(53,60,64,59)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94724);