#8943983

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	#48175	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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)(3,9)(4,13)(5,7)(6,10)(8,11)(12,15)(14,16)(17,18,21,25)(19,23,27,22)(20,28,30,24)(26,31,29,32)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(1,3,8,12)(2,9,11,15)(4,14,5,10)(6,13,16,7)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(1,4,8,5)(2,7,11,13)(3,10,12,14)(6,9,16,15)(17,19,21,27)(18,22,25,23)(20,29,30,26)(24,32,28,31)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,5,8,4)(2,7,11,13)(3,10,12,14)(6,15,16,9)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,6)(2,10)(3,7)(4,15)(5,9)(8,16)(11,14)(12,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,32)(28,31)(29,30)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,7)(2,4)(3,6)(5,11)(8,13)(9,14)(10,15)(12,16)(17,20,21,30)(18,24,25,28)(19,26,27,29)(22,31,23,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gh,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, 8851179);