#8843570

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

3830 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8750766);