#9009235

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

3923 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(33,34,40,44)(35,39,37,42)(36,46,48,41)(38,43,45,47);(2,6)(7,9)(8,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,35,40,37)(34,39,44,42)(36,38,48,45)(41,47,46,43);(2,6)(3,10)(4,13)(14,15)(18,22)(19,25)(20,27)(31,32)(33,36,40,48)(34,41,44,46)(35,45,37,38)(39,43,42,47);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,21,25)(18,23,22,29)(20,28,27,26)(24,32,30,31)(33,37,40,35)(34,42,44,39)(36,45,48,38)(41,43,46,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20,21,27)(18,24,22,30)(19,26,25,28)(23,31,29,32)(33,38,40,45)(34,43,44,47)(35,36,37,48)(39,46,42,41);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,39,40,42)(34,35,44,37)(36,47,48,43)(38,46,45,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=gh,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, 8916431);