#8228669

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

4232 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,26)(20,21)(22,23)(24,27)(25,30)(28,31)(29,32)(33,34)(35,40)(36,42)(37,45)(38,44)(39,41)(43,48)(46,47);(2,6)(7,9)(8,13)(14,15)(18,23)(19,27)(20,30)(29,31)(33,35,39,38)(34,40,41,44)(36,43,47,45)(37,42,48,46);(4,12)(8,13)(10,16)(14,15)(18,23)(21,25)(24,26)(29,31)(34,41)(35,38)(42,46)(43,45);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19,22,27)(18,24,23,26)(20,28,30,32)(21,29,25,31)(33,36,39,47)(34,42,41,46)(35,45,38,43)(37,44,48,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20)(18,25)(19,28)(21,23)(22,30)(24,31)(26,29)(27,32)(33,37,39,48)(34,43,41,45)(35,46,38,42)(36,44,47,40);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,25)(18,20,23,30)(19,29,27,31)(24,28,26,32)(33,38,39,35)(34,44,41,40)(36,43,47,45)(37,42,48,46)

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8135865);