#187512

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62752	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2005 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)(4,18,20,26)(5,10)(6,12)(7,11)(9,19,25,21)(14,28)(15,24,16,23)(22,29)(27,31)(30,32)(33,34,38,42,39,44,37,43)(35,45,47,40,48,55,53,56)(36,49,51,41,52,57,50,58)(46,60,61,64,62,54,63,59);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,32)(19,29)(20,30)(21,31)(25,27)(26,28)(33,35)(34,40)(36,46)(37,53)(38,47)(39,48)(41,54)(42,55)(43,45)(44,56)(49,64)(50,63)(51,61)(52,62)(57,59)(58,60);(1,4,7,20)(2,9,11,25)(3,14,17,30)(5,19,6,21)(8,22,13,27)(10,18,12,26)(15,29,16,31)(23,32,24,28)(33,36,39,52)(34,41,44,58)(35,46,48,62)(37,51,38,50)(40,54,56,60)(42,49,43,57)(45,59,55,64)(47,63,53,61);(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)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,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,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, 94708);