#8417207

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	#52494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(3,13)(4,6)(5,10)(7,11)(8,12)(9,16)(14,15)(17,18)(19,27)(20,31)(21,32)(22,26)(23,24)(25,28)(29,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,8,16)(2,9,12,13)(4,14,5,11)(6,7,10,15)(18,24)(19,28)(20,32)(26,30)(34,38)(39,41)(40,45)(46,47);(1,4,8,5)(2,10,12,6)(3,14,16,11)(7,9,15,13)(17,19,23,28)(18,25,24,27)(20,29,32,22)(21,30,31,26)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,8,4)(2,10,12,6)(3,11,16,14)(7,9,15,13)(17,20)(18,21)(19,29)(22,28)(23,32)(24,31)(25,30)(26,27)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,6,8,10)(2,5,12,4)(3,15,16,7)(9,14,13,11)(17,21)(18,20)(19,30)(22,27)(23,31)(24,32)(25,29)(26,28);(1,7,8,15)(2,11,12,14)(3,10,16,6)(4,13,5,9)(17,22,23,29)(18,26,24,30)(19,20,28,32)(21,27,31,25)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=g,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, 8324403);