#8734306

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34)(35,41)(36,40)(37,45)(38,42)(39,47)(43,46)(44,48);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35)(34,37)(36,43)(38,45)(39,46)(40,48)(41,42)(44,47);(2,3)(7,9)(8,10)(15,16)(19,27)(20,28)(23,32)(24,29)(33,36,38,47)(34,39,42,40)(35,43,45,44)(37,46,41,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,36)(34,40)(35,44)(37,46)(38,47)(39,42)(41,48)(43,45);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,37)(34,35)(36,46)(38,41)(39,43)(40,44)(42,45)(47,48);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,35)(34,41)(36,43)(37,42)(38,45)(39,48)(40,46)(44,47)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=gh,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, 8641502);