#8621785

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8528981);