#8746951

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

3911 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8654147);