#8621977

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

4175 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529173);