#9057305

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	#53450	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4144 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)(3,7)(4,11)(5,13)(6,8)(9,14)(10,16)(12,15)(19,25)(23,26)(27,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3,6,4)(2,7,8,11)(5,10,14,12)(9,15,13,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,34,38,42)(35,39,40,36)(37,41,45,48)(43,46,47,44);(2,8)(3,4)(5,14)(15,16)(18,22)(19,25)(20,28)(30,32)(35,40)(36,39)(37,45)(41,48);(1,4,6,3)(2,7,8,11)(5,12,14,10)(9,15,13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,46)(44,48)(45,47);(1,2)(3,7)(4,11)(5,9)(6,8)(10,15)(12,16)(13,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,38,39)(34,40,42,35)(37,44,45,46)(41,47,48,43);(1,5,6,14)(2,9,8,13)(3,10,4,12)(7,15,11,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,41,42,48)(35,43,40,47)(36,44,39,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8964501);