#8418185

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(3,8)(4,5)(6,7)(9,13)(10,16)(11,14)(12,15)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(2,7)(4,13)(8,15)(10,14)(17,18,22,21)(19,23,27,25)(20,28,30,24)(26,32,29,31)(33,35)(34,40)(36,38)(37,47)(39,44)(41,45)(42,46)(43,48);(2,7)(5,9)(8,15)(11,16)(18,21)(23,25)(24,28)(31,32)(35,44)(38,45)(40,48)(42,47);(1,3,6,12)(2,8,7,15)(4,14,13,10)(5,11,9,16)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,36)(34,37)(35,45)(38,44)(39,41)(40,42)(43,46)(47,48);(1,4)(2,9)(3,10)(5,7)(6,13)(8,11)(12,14)(15,16)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,37)(34,41)(35,42)(36,43)(38,40)(39,46)(44,47)(45,48);(1,5,6,9)(2,4,7,13)(3,11,12,16)(8,14,15,10)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,38,39,45)(34,42,43,47)(35,36,44,41)(37,40,46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325381);