#198814

order	512
nilpotency class	3
isoclinism	isoclinic class of #198807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

964 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)(4,16,18,9)(5,10)(6,20)(7,11)(8,14)(13,27,30,32)(15,23)(17,26,21,25)(19,24)(22,29,31,28)(33,34)(35,39,44,54)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57,58,64)(45,61,60,53)(48,56)(50,55)(52,62,63,59);(1,3,6,14,7,15,5,19)(2,8,10,12,11,24,20,23)(4,13,21,28,18,30,17,29)(9,22,25,27,16,31,26,32)(33,35)(34,39,41,54)(36,43)(37,45)(38,44)(40,52,55,63)(42,53,49,61)(46,57,56,64)(47,59)(48,58)(50,62)(51,60);(1,4,7,18)(2,9,11,16)(3,13,15,30)(5,17,6,21)(8,22,24,31)(10,25,20,26)(12,27,23,32)(14,28,19,29)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,54,63)(42,56,49,46)(45,62,60,59)(53,64,61,57);(1,5,7,6)(2,10,11,20)(3,14,15,19)(4,17,18,21)(8,23,24,12)(9,25,16,26)(13,28,30,29)(22,32,31,27)(34,41)(35,44)(40,55)(42,49)(43,58)(45,60)(46,56)(59,62)

Polycyclic presentation:

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

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, 106010);