#8110631

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	#60045	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4178 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8017827);