#200945

order	512
nilpotency class	3
isoclinism	isoclinic class of #200944
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	#60962	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1060 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)(3,10)(4,7)(9,14)(11,12)(13,16)(17,18)(19,27,30,41)(20,31,33,24)(21,26)(22,25)(23,29,38,34)(28,43)(32,42,35,39)(36,40)(37,44)(45,48)(46,47);(1,2,7,14,5,9,4,6)(3,8,10,16,11,15,12,13)(17,19,21,34,22,30,36,29)(18,23,26,41,25,38,40,27)(20,28,35,44,33,45,32,46)(24,37,42,48,31,47,39,43);(1,3)(2,8)(4,12)(5,11)(6,13)(7,10)(9,15)(14,16)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,31)(26,42)(27,43)(29,46)(30,45)(32,36)(34,44)(38,47)(39,40)(41,48);(3,11)(8,15)(10,12)(13,16)(18,25)(23,38)(24,31)(26,40)(27,41)(37,47)(39,42)(43,48)

Polycyclic presentation:

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

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, 108141);