#8750724

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3815 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)(4,10)(5,9)(6,12)(7,8)(13,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,8)(4,14)(9,11)(12,13)(18,22)(23,25)(24,29)(30,31)(34,38)(35,42)(40,44)(43,48);(1,3,7,5)(2,9,8,11)(4,12,14,13)(6,10,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,4,7,14)(2,10,8,15)(3,13,5,12)(6,11,16,9)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,5,7,3)(2,11,8,9)(4,12,14,13)(6,10,16,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47);(1,6,7,16)(2,12,8,13)(3,10,5,15)(4,9,14,11)

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=gi,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, 8657920);