#8749818

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,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, 8657014);