#8360926

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

4074 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)(3,8)(4,10)(5,13)(6,12)(7,9)(11,16)(14,15)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(34,40)(36,41)(42,47)(44,48);(1,3,7,6)(2,8,9,12)(4,11,15,13)(5,10,16,14)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(2,9)(3,6)(10,14)(11,13)(19,26)(22,24)(27,29)(30,32)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,4)(2,10)(3,13)(5,8)(6,11)(7,15)(9,14)(12,16)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,5,7,16)(2,11,9,13)(3,14,6,10)(4,12,15,8)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,6,7,3)(2,12,9,8)(4,11,15,13)(5,10,16,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=hi,e^2=gh,f^2=h,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, 8268122);