#8630809

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	#2688	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)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,7,6)(2,8,9,5)(4,13,15,11)(10,14,12,16)(17,19,23,27)(18,24,21,31)(20,28,30,26)(22,25,32,29)(34,35)(39,41)(40,42)(46,47);(2,9)(5,8)(10,12)(14,16)(18,21)(19,27)(25,29)(26,28)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,7,15)(2,10,9,12)(3,11,6,13)(5,14,8,16)(17,20,23,30)(18,25,21,29)(19,26,27,28)(22,31,32,24)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,8)(2,3,9,6)(4,14,15,16)(10,11,12,13)(17,21,23,18)(19,24,27,31)(20,29,30,25)(22,28,32,26)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,6,7,3)(2,8,9,5)(4,11,15,13)(10,14,12,16)(17,22)(18,26)(19,25)(20,24)(21,28)(23,32)(27,29)(30,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=h,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, 8538005);