#8734334

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

4164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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)(33,34,38,43)(35,40,46,39)(36,37,42,41)(44,47,48,45);(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)(33,34)(35,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(2,3)(7,9)(8,10)(15,16)(19,27)(20,28)(23,32)(24,29)(33,35,38,46)(34,39,43,40)(36,44,42,48)(37,45,41,47);(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)(33,35)(34,40)(36,44)(37,47)(38,46)(39,43)(41,45)(42,48);(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)(33,36)(34,41)(35,44)(37,43)(38,42)(39,47)(40,45)(46,48);(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)(33,37,38,41)(34,42,43,36)(35,45,46,47)(39,48,40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gh,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, 8641530);