#8841613

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

4098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=gi,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, 8748809);