#187575

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

1998 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)(4,17,20,25)(5,21)(6,10)(7,11)(9,19,24,18)(13,28)(14,23,15,27)(22,30)(26,29)(31,32)(33,34,38,42,39,44,37,43)(35,45,47,40,48,55,53,56)(36,49,52,41,50,58,51,57)(46,60,62,64,61,54,63,59);(1,3,7,16)(2,8,11,12)(4,13,20,31)(5,14,6,15)(9,22,24,26)(10,23,21,27)(17,32,25,28)(18,29,19,30)(33,35,39,48)(34,40,44,56)(36,46,50,61)(37,53,38,47)(41,54,57,60)(42,55,43,45)(49,64,58,59)(51,63,52,62);(1,4)(2,9)(3,13)(5,19)(6,18)(7,20)(8,22)(10,25)(11,24)(12,26)(14,30)(15,29)(16,31)(17,21)(23,28)(27,32)(33,36)(34,41)(35,46)(37,52)(38,51)(39,50)(40,54)(42,49)(43,58)(44,57)(45,59)(47,63)(48,61)(53,62)(55,64)(56,60);(1,5,7,6)(2,10,11,21)(3,14,16,15)(4,18,20,19)(8,23,12,27)(9,17,24,25)(13,29,31,30)(22,32,26,28)(36,50)(41,57)(46,61)(49,58)(51,52)(54,60)(59,64)(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=dhi,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, 94771);