#9222117

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

4123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9129313);