#8934852

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	#47695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842048);