#8324478

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,e^2=gh,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, 8231674);