#198800

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	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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30)(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);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,17,28,16,27)(8,21,24,25,15,31,23,32)(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);(1,4,6,17)(2,8,10,15)(3,12,14,28)(5,19,20,16)(7,21,22,31)(9,24,18,23)(11,25,26,32)(13,29,30,27)(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);(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,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, 105996);