#8285536

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

4055 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8192732);