#8739429

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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)(3,5)(4,11)(6,7)(8,13)(9,15)(10,12)(14,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(2,7)(3,9)(8,11)(10,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(34,39)(35,40)(41,46)(42,47);(2,7)(3,9)(8,11)(10,16)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 8646625);