#8739385

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3814 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=gh,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, 8646581);