#8963625

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

4066 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870821);