#8418382

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	#42541	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325578);