#8901592

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

4127 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)(19,24)(20,26)(21,27)(22,31)(23,25)(28,32)(29,30)(33,34)(35,44)(36,39)(37,47)(38,42)(40,43)(41,46)(45,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,23,22)(18,24,25,31)(20,28,30,27)(21,26,32,29)(33,35)(34,41)(36,45)(37,42)(38,39)(40,46)(43,44)(47,48);(2,6)(7,13)(8,14)(15,16)(18,25)(19,22)(26,29)(27,28)(33,36,40,37)(34,39,43,47)(35,45,46,42)(38,44,48,41);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,20)(18,26)(19,28)(21,31)(22,27)(23,30)(24,32)(25,29)(33,37,40,36)(34,39,43,47)(35,42,46,45)(38,44,48,41);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,21,23,32)(18,27,25,28)(19,29,22,26)(20,24,30,31)(33,38)(34,42)(35,39)(36,44)(37,41)(40,48)(43,45)(46,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,22,23,19)(18,24,25,31)(20,27,30,28)(21,26,32,29)(33,39)(34,36)(35,38)(37,43)(40,47)(41,45)(42,44)(46,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=ghi,e^2=i,f^2=h,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, 8808788);