#8994244

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

4176 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,16)(13,14)(17,20)(18,19)(21,29)(22,26)(23,25)(24,31)(27,32)(28,30)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,21,22,30)(18,24,25,32)(19,27,23,31)(20,28,26,29)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=i,e^2=gh,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, 8901440);