#9863939

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

4126 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)(15,16)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,47)(43,44)(46,48);(1,2,6,3)(4,9,12,7)(5,8,14,10)(11,16,13,15)(17,19,23,21)(18,24,22,26)(20,27,30,29)(25,31,28,32)(33,35,38,40)(34,39,42,36)(37,43,47,48)(41,46,45,44);(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,34,38,42)(35,36,40,39)(37,41,47,45)(43,44,48,46);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32)(33,36,38,39)(34,40,42,35)(37,46,47,44)(41,43,45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,37)(34,41)(35,43)(36,44)(38,47)(39,46)(40,48)(42,45)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9771135);