#8295582

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

4190 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)(17,18,23,25)(19,26,21,24)(20,29,31,22)(27,32,30,28)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19,23,21)(18,24,25,26)(20,30,31,27)(22,28,29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(4,12)(7,14)(10,16)(13,15)(18,25)(22,29)(24,26)(28,32)(34,38)(39,41)(40,44)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,22)(19,27)(21,30)(23,31)(24,32)(25,29)(26,28)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21,23,19)(18,26,25,24)(20,27,31,30)(22,32,29,28)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,22)(18,20)(19,28)(21,32)(23,29)(24,30)(25,31)(26,27)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202778);