#8994536

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

4229 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)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,11)(12,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,31)(29,30)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(2,9)(3,5)(4,14)(6,16)(19,26)(20,28)(21,24)(25,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,4,7,14)(2,10,9,15)(3,13,5,12)(6,11,16,8)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(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)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,6)(2,12)(3,10)(4,11)(5,15)(7,16)(8,14)(9,13)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=gi,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, 8901732);