#9053965

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=i,f^2=g,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, 8961161);