#8418652

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3)(2,5)(4,12)(6,8)(7,16)(9,10)(11,15)(13,14)(18,19)(23,25)(24,26)(30,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(3,8)(5,10)(11,12)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,2)(3,10)(4,13)(5,8)(6,9)(7,15)(11,14)(12,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,4)(2,7)(3,11)(5,14)(6,15)(8,12)(9,13)(10,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47);(1,5,6,10)(2,8,9,3)(4,14,15,16)(7,12,13,11)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,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, 8325848);