#8741911

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

4118 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8649107);