#8472810

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	#50103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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)(3,10,5,7)(4,12,14,8)(11,15,13,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,36)(37,40)(38,45)(39,41)(42,47)(43,48)(44,46);(1,3,6,5)(2,7,9,10)(4,13,14,11)(8,16,12,15)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35,40,39)(34,41,37,36)(38,43,47,46)(42,48,45,44);(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36)(34,39)(35,37)(38,44)(40,41)(42,46)(43,45)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,37,40,34)(35,41,39,36)(38,45,47,42)(43,48,46,44);(1,4,6,14)(2,8,9,12)(3,11,5,13)(7,15,10,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,39,40,35)(34,41,37,36)(38,46,47,43)(42,48,45,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8380006);