#8323352

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

4074 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230548);