#8417246

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	#52359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(3,6)(4,10)(5,11)(7,9)(8,12)(13,14)(15,16)(18,22)(23,28)(24,29)(30,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3,8,7)(2,9,12,6)(4,13,5,15)(10,16,11,14)(19,25)(23,28)(26,31)(30,32)(34,39)(35,40)(41,46)(42,47);(1,4,8,5)(2,10,12,11)(3,13,7,15)(6,14,9,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,5,8,4)(2,11,12,10)(3,13,7,15)(6,14,9,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,6,8,9)(2,3,12,7)(4,14,5,16)(10,13,11,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,7,8,3)(2,9,12,6)(4,13,5,15)(10,14,11,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=gi,e^2=gi,f^2=gh,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, 8324442);