#8371528

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	#42608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4122 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(34,40)(36,41)(42,47)(44,48);(2,7)(3,10)(4,13)(15,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(2,7)(5,8)(9,12)(15,16)(18,23)(21,25)(24,31)(27,32)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,3)(2,8)(4,14)(5,7)(6,10)(9,15)(11,13)(12,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,5,6,8)(2,3,7,10)(4,15,13,16)(9,11,12,14)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8278724);