#8734321

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

4189 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,30)(27,32)(29,31)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(2,7)(3,8)(9,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(2,7)(3,8)(9,16)(10,13)(18,19)(23,25)(24,26)(31,32)(36,44)(40,47)(43,45)(46,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37,38,35)(34,39,41,42)(36,43,44,45)(40,48,47,46)

Polycyclic presentation:

2^9|b^2=ghi,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, 8641517);