#8348938

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

4255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8256134);