#8633973

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	#45468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=h,f^2=hi,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, 8541169);