#8963662

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

4231 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=gi,e^2=gi,f^2=g,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, 8870858);