#8406485

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

3975 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8313681);