#9490183

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

4147 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(33,34,39,43)(35,42,47,40)(36,41,37,38)(44,46,45,48);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,7)(3,10)(8,15)(11,12)(18,23)(20,27)(21,29)(26,31)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,48,42,46);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,35)(34,42)(36,45)(37,44)(38,46)(39,47)(40,43)(41,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9397379);