#200998

order	512
nilpotency class	3
isoclinism	isoclinic class of #200932
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	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(3,9)(4,12)(8,14)(10,11)(13,15)(17,18,22,28,23,26,21,27)(19,24,32,40,30,38,31,39)(20,33,36,25,34,42,35,41)(29,43,45,37,46,47,44,48);(1,2,4,6,5,8,12,14)(3,7,10,13,11,16,9,15)(17,19,22,31,23,30,21,32)(18,24,27,40,26,38,28,39)(20,29,35,45,34,46,36,44)(25,37,42,43,41,48,33,47);(1,3)(2,7)(4,10)(5,11)(6,13)(8,16)(9,12)(14,15)(17,20,23,34)(18,25,26,41)(19,29,30,46)(21,36,22,35)(24,37,38,48)(27,42,28,33)(31,45,32,44)(39,47,40,43);(18,26)(24,38)(25,41)(27,28)(33,42)(37,48)(39,40)(43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108194);