#8581448

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8488644);