#8806732

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

4118 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)(3,12)(4,13)(5,11)(6,16)(7,9)(8,14)(10,15)(17,18,22,27)(19,26,30,23)(20,29,31,24)(21,25,32,28)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(1,3,7,14)(2,8,9,12)(4,5,15,16)(6,10,11,13)(17,19,22,30)(18,23,27,26)(20,28,31,25)(21,29,32,24)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,9)(4,15)(5,16)(8,12)(20,31)(21,32)(24,29)(25,28)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,4)(2,10)(3,5)(6,12)(7,15)(8,11)(9,13)(14,16)(17,20,22,31)(18,24,27,29)(19,28,30,25)(21,26,32,23)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,5)(2,11)(3,4)(6,9)(7,16)(8,10)(12,13)(14,15)(17,21,22,32)(18,25,27,28)(19,29,30,24)(20,26,31,23)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,6,7,11)(2,5,9,16)(3,13,14,10)(4,12,15,8)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8713928);