#8634140

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	#46042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541336);