#8943912

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	#40852	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8851108);