#8838374

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

4154 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,38,39)(35,40,46,41)(36,37,43,42)(44,45,47,48);(2,7)(4,8)(5,14)(11,16)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(34,39)(37,42)(40,41)(45,48);(2,7)(3,9)(10,13)(11,16)(19,27)(20,29)(23,28)(24,32)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,13)(14,16)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,36,38,43)(34,42,39,37)(35,44,46,47)(40,48,41,45);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8745570);