#8372180

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

4216 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=gh,f^2=gi,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, 8279376);