#8876378

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

4241 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)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18,22,23)(19,25,26,24)(20,29,31,21)(27,32,30,28)(35,41)(39,42)(43,47)(46,48);(2,7)(4,8)(5,14)(11,16)(18,23)(21,29)(24,25)(28,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(2,7)(3,9)(10,13)(11,16)(19,26)(24,25)(27,30)(28,32)(34,38)(39,42)(40,44)(46,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,19,22,26)(18,24,23,25)(20,30,31,27)(21,28,29,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,20)(18,21)(19,27)(22,31)(23,29)(24,32)(25,28)(26,30)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21)(18,20)(19,28)(22,29)(23,31)(24,30)(25,27)(26,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

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

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, 8783574);