#8107349

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	#52475	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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)(3,5)(4,12)(6,7)(8,14)(9,10)(11,16)(13,15)(17,18)(19,26)(20,21)(22,25)(23,29)(24,30)(27,31)(28,32)(34,38)(39,44)(40,45)(46,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(35,41)(39,44)(42,47)(46,48);(2,7)(3,9)(8,12)(11,15)(17,20,22,30)(18,21,25,24)(19,27,29,32)(23,28,26,31)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,14)(13,15)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,28)(29,32)(33,35,37,41)(34,39,38,44)(36,42,43,47)(40,46,45,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,12)(9,15)(10,16)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,26,32)(33,34)(35,39)(36,40)(37,38)(41,44)(42,46)(43,45)(47,48);(1,5,6,10)(2,9,7,3)(4,13,14,16)(8,15,12,11)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014545);