#8838438

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

4073 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)(3,5)(4,12)(6,7)(8,10)(9,13)(11,16)(14,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,28)(29,32)(33,34,38,41)(35,37,42,39)(36,40,46,48)(43,45,44,47);(2,7)(4,13)(5,8)(11,14)(17,19)(18,23)(20,21)(22,27)(24,30)(25,32)(26,28)(29,31)(35,42)(37,39)(43,44)(45,47);(2,7)(3,10)(9,12)(11,14)(19,27)(20,29)(23,28)(24,32)(33,35)(34,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,3,6,10)(2,8,7,5)(4,14,13,11)(9,15,12,16)(17,20,22,29)(18,24,26,32)(19,21,27,31)(23,30,28,25)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4,6,13)(2,9,7,12)(3,11,10,14)(5,15,8,16)(17,21)(18,25)(19,20)(22,31)(23,32)(24,28)(26,30)(27,29)(33,35)(34,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,5,6,8)(2,10,7,3)(4,15,13,16)(9,14,12,11)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,32)(30,31)(33,37,38,39)(34,35,41,42)(36,45,46,47)(40,43,48,44)

Polycyclic presentation:

2^9|a^2=gh,d^2=hi,e^2=g,f^2=h,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, 8745634);