#8824453

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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)(4,9)(6,7)(8,11)(10,13)(12,16)(14,15)(18,22)(23,28)(24,29)(30,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,7)(5,8)(9,15)(13,16)(20,27)(24,29)(25,31)(30,32)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(1,3,6,11)(2,5,7,8)(4,12,14,10)(9,16,15,13)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,16)(12,14)(13,15)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4,6,14)(2,9,7,15)(3,10,11,12)(5,13,8,16)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,8)(2,3,7,11)(4,13,14,16)(9,10,15,12)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731649);