#8824438

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

4139 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)(3,8,11,5)(4,9,14,15)(10,16,12,13)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(2,7)(5,8)(9,15)(13,16)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(37,45)(40,44)(43,47)(46,48);(1,3,6,11)(2,8,7,5)(4,12,14,10)(9,13,15,16)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,3)(2,5)(4,10)(6,11)(7,8)(9,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,4,6,14)(2,9,7,15)(3,10,11,12)(5,13,8,16)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,5)(2,3)(4,13)(6,8)(7,11)(9,10)(12,15)(14,16)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731634);