#8922533

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	#50927	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=i,e^2=gi,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, 8829729);