#8862440

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

4094 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=gi,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, 8769636);