#8901582

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

4120 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808778);