#9970147

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

4190 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)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48);(4,11)(8,14)(10,15)(12,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,43)(44,46)(47,48);(2,7)(3,9)(8,14)(10,15)(18,23)(19,26)(24,29)(27,32)(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)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(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)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(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)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)(33,37)(34,42)(35,41)(36,39)(38,47)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,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, 9877343);