#8324291

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

4228 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=h,e^2=g,f^2=hi,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, 8231487);