#8678973

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8586169);