#8734670

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	#52639	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gh,f^2=g,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, 8641866);