#8761960

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

4200 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669156);