#8917834

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3862 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825030);