#9970150

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

4174 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(4,11)(8,14)(10,15)(12,16)(17,19,22,24)(18,21,25,26)(20,27,28,31)(23,29,30,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(2,7)(3,9)(8,14)(10,15)(20,28)(23,30)(27,31)(29,32)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,30)(26,32)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,26)(18,24,25,19)(20,29,28,32)(23,31,30,27)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,30)(31,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^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, 9877346);