#8997166

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

4063 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8904362);