#8159976

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

4125 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)(3,6)(4,9)(5,10)(7,12)(8,13)(11,14)(15,16)(17,18,22,24)(19,23,25,20)(21,29,30,26)(27,28,32,31)(37,45)(40,46)(42,47)(43,48);(2,3)(7,9)(8,10)(14,15)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,46)(47,48);(2,3)(7,9)(8,10)(14,15)(18,24)(19,25)(26,29)(27,32)(34,39)(35,41)(40,46)(42,47);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19)(18,23)(20,24)(21,27)(22,25)(26,31)(28,29)(30,32)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8067172);