#9040668

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

3978 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)(3,4)(5,9)(6,14)(7,8)(10,15)(11,16)(12,13)(33,34,39,37)(35,38,43,40)(36,44,47,41)(42,48,45,46);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35,39,43)(34,40,37,38)(36,45,47,42)(41,46,44,48);(2,8)(3,5)(6,15)(12,16)(18,22)(20,29)(23,25)(26,30)(34,37)(38,40)(41,44)(46,48);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,36,39,47)(34,41,37,44)(35,42,43,45)(38,46,40,48);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37,39,34)(35,38,43,40)(36,41,47,44)(42,48,45,46);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,38)(34,35)(36,46)(37,43)(39,40)(41,42)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947864);