#9007828

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	#40292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8915024);