#8107585

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

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, 8014781);