#8902485

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8809681);