#8107210

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	#52388	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,16)(12,15)(17,18)(19,26)(20,21)(22,24)(23,27)(25,31)(28,32)(29,30)(34,37)(35,41)(36,44)(46,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19,23,22)(18,24,27,26)(20,28,31,30)(21,29,25,32)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(2,7)(3,8)(9,13)(11,15)(17,20,23,31)(18,21,27,25)(19,28,22,30)(24,29,26,32)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,20)(18,25)(19,28)(21,27)(22,30)(23,31)(24,32)(26,29)(33,35,38,41)(34,39,37,42)(36,43,44,47)(40,46,45,48);(1,4,6,10)(2,8,7,3)(5,12,14,16)(9,15,13,11)(17,21,23,25)(18,20,27,31)(19,29,22,32)(24,28,26,30)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,19)(18,26,27,24)(20,30,31,28)(21,32,25,29)(33,37,38,34)(35,42,41,39)(36,45,44,40)(43,48,47,46)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=hi,e^2=gh,f^2=g,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, 8014406);