#9910139

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

4053 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9817335);