#8348933

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^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,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, 8256129);