#8322246

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229442);