#198806

order	512
nilpotency class	3
isoclinism	isoclinic class of #198798
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 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)(4,17,19,9)(5,10)(6,20)(7,11)(8,15)(13,26,29,32)(14,23)(16,27)(18,25,21,24)(22,28,31,30)(33,34)(35,39,44,53)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57,58,64)(45,61,60,54)(48,56)(50,55)(52,62,63,59);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,20,27)(4,13,21,30,19,29,18,28)(9,22,24,26,17,31,25,32)(33,35)(34,39,41,53)(36,43)(37,45)(38,44)(40,52,55,63)(42,54,49,61)(46,57,56,64)(47,59)(48,58)(50,62)(51,60);(1,4,7,19)(2,9,11,17)(3,13,16,29)(5,18,6,21)(8,22,23,31)(10,24,20,25)(12,26,27,32)(14,28,15,30)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(42,56,49,46)(45,62,60,59)(54,64,61,57);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,12,23,27)(9,24,17,25)(13,28,29,30)(22,26,31,32)(34,41)(39,53)(40,55)(42,49)(46,56)(52,63)(54,61)(57,64)

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,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, 106002);