#8718962

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	#44066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4136 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)(3,10)(4,12)(5,8)(6,9)(7,13)(11,15)(14,16)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,31)(29,30)(33,34)(35,44)(36,46)(37,45)(38,41)(39,40)(42,47)(43,48);(1,3,6,8)(2,5,9,10)(4,11,13,16)(7,15,12,14)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,32,31)(34,40)(36,47)(41,44)(43,45);(4,13)(7,12)(11,16)(14,15)(18,25)(19,27)(20,26)(29,31)(33,35,39,38)(34,41,40,44)(36,43,47,45)(37,42,48,46);(1,4,6,13)(2,7,9,12)(3,11,8,16)(5,15,10,14)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,29,24,31)(33,36,39,47)(34,42,40,46)(35,45,38,43)(37,44,48,41);(1,2)(3,5)(4,7)(6,9)(8,10)(11,15)(12,13)(14,16)(17,21)(18,26)(19,29)(20,25)(22,30)(23,32)(24,28)(27,31)(33,37)(34,43)(35,42)(36,44)(38,46)(39,48)(40,45)(41,47);(1,5)(2,8)(3,9)(4,14)(6,10)(7,11)(12,16)(13,15)(17,22,23,24)(18,27,25,19)(20,31,26,29)(21,28,32,30)(33,38,39,35)(34,41,40,44)(36,45,47,43)(37,42,48,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=gi,f^2=h,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, 8626158);