#8745839

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,36)(40,43)(41,42)(47,48);(2,7)(3,9)(8,11)(10,16)(19,25)(22,28)(26,31)(29,32)(33,34,39,36)(35,43,46,40)(37,41,38,42)(44,48,45,47);(2,7)(3,9)(8,11)(10,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,46)(34,40,36,43)(37,44,38,45)(41,47,42,48);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36,39,34)(35,43,46,40)(37,42,38,41)(44,48,45,47);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,38)(34,41,36,42)(35,44,46,45)(40,47,43,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,38,39,37)(34,42,36,41)(35,45,46,44)(40,48,43,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,d^2=hi,e^2=g,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, 8653035);