#8402297

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

4153 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)(3,5)(4,9)(6,11)(7,8)(10,15)(12,14)(13,16)(34,35)(36,43)(40,42)(44,48);(1,3,7,10)(2,5,8,15)(4,12,13,6)(9,14,16,11)(17,18)(19,26)(20,21)(22,23)(24,29)(25,31)(27,28)(30,32)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(2,8)(3,10)(6,12)(9,16)(18,23)(21,25)(24,26)(28,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(1,4)(2,9)(3,12)(5,14)(6,10)(7,13)(8,16)(11,15)(17,19,22,29)(18,24,23,26)(20,30,31,27)(21,28,25,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,5,7,15)(2,10,8,3)(4,11,13,14)(6,9,12,16)(17,20)(18,25)(19,27)(21,23)(22,31)(24,28)(26,32)(29,30)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,6,7,12)(2,11,8,14)(3,4,10,13)(5,9,15,16)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,30,26,27)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309493);