#8962933

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

4109 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8870129);