#8720815

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

4007 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)(3,11)(4,13)(5,8)(6,12)(7,9)(10,15)(14,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,30)(31,32)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,10,16,13)(17,18)(19,26)(20,24)(21,25)(22,23)(27,30)(28,29)(31,32)(34,40)(35,36)(42,45)(43,44);(2,9)(4,15)(8,11)(12,14)(18,23)(19,24)(25,32)(27,29)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,4)(2,10)(3,12)(5,14)(6,8)(7,15)(9,13)(11,16)(17,19,22,24)(18,20,23,26)(21,29,31,27)(25,30,32,28)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,10,16,13)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,6,7,16)(2,12,9,14)(3,10,5,13)(4,11,15,8)(17,21,22,31)(18,25,23,32)(19,27,24,29)(20,28,26,30)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628011);