#8973367

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8880563);