#9970129

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

4360 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dghi,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9877325);