#9175008

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

4110 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)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,18,23,25)(19,27,28,24)(20,22,29,30)(21,26,31,32)(34,39)(35,41)(37,45)(44,46);(1,3,6,10)(2,7,9,4)(5,11,14,16)(8,15,13,12)(17,19,23,28)(18,24,25,27)(20,26,29,32)(21,30,31,22)(34,39)(36,42)(40,47)(44,46);(3,10)(4,7)(11,16)(12,15)(18,25)(20,29)(24,27)(26,32)(33,34,38,39)(35,42,41,36)(37,40,45,47)(43,46,48,44);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,20)(18,22)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,35,38,41)(34,36,39,42)(37,43,45,48)(40,44,47,46);(1,2)(3,7)(4,10)(5,13)(6,9)(8,14)(11,12)(15,16)(17,21,23,31)(18,26,25,32)(19,22,28,30)(20,27,29,24)(33,36)(34,35)(37,46)(38,42)(39,41)(40,48)(43,47)(44,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,20)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9082204);