#8749701

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

3887 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gh,e^2=h,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, 8656897);