#8726519

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(33,34,39,42)(35,38,40,36)(37,41,47,48)(43,46,45,44);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,38)(36,42)(37,45)(39,40)(41,44)(43,47)(46,48);(2,9)(4,15)(8,11)(12,14)(17,18,21,24)(19,22,26,31)(20,23,29,27)(25,30,28,32)(33,36,39,38)(34,40,42,35)(37,46,47,44)(41,43,48,45);(1,4,7,15)(2,10,9,6)(3,12,5,14)(8,13,11,16)(17,18)(19,22)(20,27)(21,24)(23,29)(25,32)(26,31)(28,30)(33,37,39,47)(34,41,42,48)(35,43,40,45)(36,44,38,46);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,13,10,16)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,38,39,36)(34,40,42,35)(37,44,47,46)(41,43,48,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633715);