#8618156

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,e^2=gi,f^2=g,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, 8525352);