#8734597

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

4103 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)(3,11)(4,5)(6,8)(7,9)(10,15)(12,13)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,43)(35,41,44,39)(36,40,45,46)(37,42,47,48);(1,3,7,6)(2,8,9,11)(4,14,10,12)(5,13,15,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35)(34,39)(36,42)(37,40)(38,44)(41,43)(45,48)(46,47);(2,9)(4,10)(8,11)(12,14)(19,27)(20,28)(23,32)(24,29)(33,36,38,45)(34,40,43,46)(35,42,44,48)(37,41,47,39);(1,4,7,10)(2,5,9,15)(3,12,6,14)(8,16,11,13)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36)(34,40)(35,42)(37,39)(38,45)(41,47)(43,46)(44,48);(1,5)(2,10)(3,13)(4,9)(6,16)(7,15)(8,12)(11,14)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,35)(34,41)(36,42)(37,46)(38,44)(39,43)(40,47)(45,48);(1,6,7,3)(2,8,9,11)(4,12,10,14)(5,13,15,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37,38,47)(34,42,43,48)(35,40,44,46)(36,41,45,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8641793);