#8986977

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	#41858	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8894173);