#8602483

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	#49219	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18)(19,20)(21,29)(22,26)(23,24)(25,32)(27,30)(28,31)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(2,7)(3,11)(4,14)(13,15)(17,19,23,27)(18,20,24,30)(21,31,32,22)(25,26,29,28)(34,40)(35,36)(42,47)(43,44);(2,7)(4,14)(8,10)(12,16)(18,24)(19,27)(21,32)(26,28)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,15,9,13)(17,20)(18,19)(21,28)(22,25)(23,30)(24,27)(26,32)(29,31)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,21)(18,25)(19,22)(20,28)(23,32)(24,29)(26,30)(27,31)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,22)(18,26)(19,21)(20,29)(23,31)(24,28)(25,30)(27,32)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509679);