#8839185

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

4260 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)(4,11)(5,13)(6,7)(8,15)(10,12)(14,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34)(35,40)(36,46)(37,42)(38,39)(41,45)(43,48)(44,47);(1,3,6,11)(2,4,7,9)(5,14,15,10)(8,16,13,12)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(34,39)(37,46)(40,41)(44,48);(2,7)(4,9)(8,13)(12,16)(19,27)(20,29)(23,28)(24,32)(33,35,38,45)(34,40,39,41)(36,43,42,47)(37,44,46,48);(1,4,6,9)(2,3,7,11)(5,12,15,16)(8,10,13,14)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,35)(34,41)(36,47)(37,44)(38,45)(39,40)(42,43)(46,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,36,38,42)(34,37,39,46)(35,43,45,47)(40,44,41,48);(1,5,6,15)(2,8,7,13)(3,10,11,14)(4,12,9,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,37)(34,42)(35,44)(36,39)(38,46)(40,47)(41,43)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8746381);