#8824439

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	#48057	C₂
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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(34,37)(39,42)(40,45)(46,48);(2,6)(7,9)(8,13)(14,15)(17,19,21,25)(18,22,23,31)(20,28,29,26)(24,32,27,30)(36,44)(40,45)(41,47)(46,48);(3,10)(4,12)(7,9)(8,13)(18,23)(19,25)(24,27)(26,28)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23)(25,31)(26,30)(27,29)(28,32)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,19)(18,22)(20,26)(21,25)(23,31)(24,30)(27,32)(28,29)(33,36,38,44)(34,40,37,45)(35,41,43,47)(39,46,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,27)(25,28)(31,32)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731635);