#8824785

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

4141 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=g,f^2=g,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, 8731981);