#8922864

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	#58060	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8830060);