#8907954

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

4259 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(34,39)(35,43)(41,44)(42,45);(1,3)(2,8)(4,12)(5,14)(6,13)(7,11)(9,16)(10,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,34,38,39)(35,40,43,46)(36,41,47,44)(37,42,48,45);(2,6)(3,11)(4,14)(10,16)(19,27)(20,29)(23,30)(24,32)(35,43)(37,48)(40,46)(42,45);(1,4,7,14)(2,9,6,15)(3,12,11,5)(8,16,13,10)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,35,38,43)(34,40,39,46)(36,45,47,42)(37,44,48,41);(1,5,7,12)(2,10,6,16)(3,4,11,14)(8,9,13,15)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,36)(34,41)(35,42)(37,46)(38,47)(39,44)(40,48)(43,45);(1,6,7,2)(3,13,11,8)(4,15,14,9)(5,16,12,10)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,37,38,48)(34,42,39,45)(35,44,43,41)(36,46,47,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815150);