#8922596

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

4125 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)(3,8,12,7)(4,13,14,5)(10,11,15,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,16)(14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(34,39)(36,42)(41,45)(44,48);(4,14)(5,13)(10,15)(11,16)(19,25)(20,28)(23,27)(24,29)(33,35,38,40)(34,36,39,42)(37,46,47,43)(41,48,45,44);(1,4)(2,5)(3,10)(6,14)(7,11)(8,16)(9,13)(12,15)(17,20,22,28)(18,24,21,29)(19,26,25,31)(23,30,27,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47);(1,3)(2,8)(4,15)(5,11)(6,12)(7,9)(10,14)(13,16)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,46)(43,45)(47,48);(1,5)(2,4)(3,11)(6,13)(7,10)(8,15)(9,14)(12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,37,38,47)(34,41,39,45)(35,43,40,46)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829792);