#187725

order	512
nilpotency class	3
isoclinism	isoclinic class of #187721
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1981 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)(5,12)(6,10)(7,11)(9,19)(14,28,30,32)(15,23,16,24)(20,26)(21,25)(22,29,27,31)(33,34)(35,43,46,57)(36,40)(37,42)(38,41)(39,45,53,49)(44,60,62,59)(47,55)(48,54)(50,58)(51,56)(52,61,64,63);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,22)(10,24)(11,13)(12,23)(18,28)(19,31)(20,30)(21,29)(25,27)(26,32)(33,35)(34,39)(36,44)(37,49)(38,46)(40,52)(41,53)(42,57)(43,56)(45,51)(47,63)(48,62)(50,61)(54,64)(55,59)(58,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,25)(12,26)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28)(33,36,38,48)(34,40,41,54)(35,44,46,62)(37,50,51,47)(39,52,53,64)(42,58,56,55)(43,59,57,60)(45,63,49,61);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,20,21)(8,23,13,24)(9,18,25,26)(14,29,30,31)(22,32,27,28)(34,41)(39,53)(40,54)(42,56)(43,57)(52,64)(55,58)(59,60)

Polycyclic presentation:

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

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, 94921);