#8115429

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	#42504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022625);