#8416803

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	#53096	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gi,e^2=gi,f^2=h,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, 8323999);