#8118620

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

4041 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025816);