#8733785

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	#52379	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8640981);