#8944063

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=h,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, 8851259);