#8383780

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

4047 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=i,e^2=g,f^2=ghi,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, 8290976);