#8534443

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	#52316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4267 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441639);