#8841607

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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)(19,24)(21,26)(22,30)(25,28)(34,40)(35,45)(36,38)(43,48);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,23,20)(19,26,24,21)(22,28,30,25)(27,29,31,32)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(2,3)(7,9)(8,10)(14,15)(17,19,23,24)(18,21,20,26)(22,27,30,31)(25,29,28,32)(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)(17,20,23,18)(19,21,24,26)(22,28,30,25)(27,29,31,32)(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)(18,24)(19,20)(22,29)(23,26)(25,31)(27,28)(30,32)(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)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=g,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, 8748803);