#8747932

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,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, 8655128);