#9506875

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

4174 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=i,f^2=g,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, 9414071);