#8898558

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

4178 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=h,e^2=g,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, 8805754);