#8230698

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8137894);