#8285450

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

4129 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)(17,18,22,23)(19,25,27,21)(20,28,31,24)(26,30,29,32)(33,34,39,42)(35,40,43,38)(36,37,41,47)(44,45,48,46);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,21)(20,29)(22,27)(23,25)(24,32)(26,31)(28,30)(33,35,39,43)(34,40,42,38)(36,44,41,48)(37,45,47,46);(4,12)(7,14)(10,16)(13,15)(18,23)(21,25)(24,28)(30,32)(35,43)(38,40)(44,48)(45,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20,22,31)(18,24,23,28)(19,26,27,29)(21,30,25,32)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,21)(20,29)(22,27)(23,25)(24,32)(26,31)(28,30)(33,37)(34,41)(35,45)(36,42)(38,44)(39,47)(40,48)(43,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,29)(31,32)(33,38)(34,35)(36,46)(37,44)(39,40)(41,45)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=gh,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, 8192646);