#9015525

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	#44236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,d^2=hi,e^2=i,f^2=h,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, 8922721);