#9815111

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=hi,f^2=gh,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, 9722307);