#9970135

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

4180 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)(3,7)(4,9)(5,10)(6,11)(8,13)(12,15)(14,16)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(1,3)(2,7)(4,10)(5,9)(6,13)(8,11)(12,14)(15,16)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,3,6,13)(2,7,11,8)(4,12,14,10)(5,9,15,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(1,3)(2,8)(4,12)(5,9)(6,13)(7,11)(10,14)(15,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4)(2,9)(3,12)(5,8)(6,14)(7,15)(10,13)(11,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,34)(35,40,36,42)(38,41,46,45)(43,48,44,47);(1,5)(2,10)(3,9)(4,7)(6,15)(8,14)(11,12)(13,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9877331);