#8994530

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

4242 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)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(18,23)(19,26)(25,31)(27,29)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,11)(12,14)(15,16)(17,18,22,23)(19,24,26,21)(20,25,28,31)(27,32,29,30)(34,38)(39,41)(40,44)(47,48);(2,9)(3,5)(4,14)(6,16)(19,26)(20,28)(21,24)(25,31)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,19)(18,24)(20,29)(21,23)(22,26)(25,30)(27,28)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,5,7,3)(2,11,9,8)(4,13,14,12)(6,15,16,10)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,21,22,24)(18,19,23,26)(20,30,28,32)(25,27,31,29)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8901726);