#8621616

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(17,18)(19,24)(20,21)(22,27)(23,25)(26,31)(28,32)(29,30)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(1,3,7,6)(2,8,9,5)(4,13,15,11)(10,14,12,16)(17,19,23,28)(18,24,25,32)(20,29,26,27)(21,30,31,22)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,9)(5,8)(10,12)(14,16)(18,25)(19,28)(21,31)(27,29)(35,40)(37,45)(43,46)(44,48);(1,4,7,15)(2,10,9,12)(3,11,6,13)(5,14,8,16)(17,20,23,26)(18,21,25,31)(19,27,28,29)(22,32,30,24)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,5,7,8)(2,3,9,6)(4,14,15,16)(10,11,12,13)(17,21)(18,26)(19,22)(20,25)(23,31)(24,29)(27,32)(28,30)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,6,7,3)(2,8,9,5)(4,11,15,13)(10,14,12,16)(17,22)(18,27)(19,21)(20,24)(23,30)(25,29)(26,32)(28,31)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=h,e^2=hi,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, 8528812);