#8107661

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	#42068	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=hi,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, 8014857);