#9009228

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

3994 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,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, 8916424);