#8393703

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	#46052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300899);