#8351624

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

4191 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,38,40)(35,39,43,47)(36,41,46,45)(37,44,48,42);(2,7)(4,8)(9,14)(12,15)(17,19,22,27)(18,23,21,25)(20,29,30,26)(24,32,28,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(5,13)(9,14)(10,16)(12,15)(18,21)(23,25)(24,28)(31,32)(34,40)(35,43)(41,45)(42,44);(1,3,6,11)(2,8,7,4)(5,10,13,16)(9,15,14,12)(17,20,22,30)(18,24,21,28)(19,26,27,29)(23,31,25,32)(33,36)(34,41)(35,44)(37,47)(38,46)(39,48)(40,45)(42,43);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,37,38,48)(34,42,40,44)(35,45,43,41)(36,39,46,47);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,35)(34,39)(36,44)(37,45)(38,43)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8258820);