#8115332

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	#37659	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4541 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022528);