#8963722

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	#50114	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=hi,e^2=i,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, 8870918);