#9970059

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

4378 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(3,9)(6,13)(10,15)(14,16)(18,20)(19,26)(25,28)(27,32)(33,35,38,45)(34,39,43,40)(36,44,46,42)(37,41,47,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,35)(34,40)(36,44)(37,41)(38,45)(39,43)(42,46)(47,48);(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,36)(34,41)(35,44)(37,40)(38,46)(39,47)(42,45)(43,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^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, 9877255);