#8994280

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
prime radical	#43421	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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)(3,4)(5,13)(6,8)(7,9)(10,14)(11,15)(12,16)(18,23)(19,26)(25,31)(27,29);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,31)(29,30)(33,34,38,42)(35,40,45,39)(36,37,41,48)(43,47,46,44);(2,8)(4,7)(5,14)(11,16)(19,26)(20,28)(21,24)(25,31)(33,35,38,45)(34,39,42,40)(36,46,41,43)(37,47,48,44);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,14)(15,16)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(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=i,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, 8901476);