#10143673

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050869);