#8908347

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

4008 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)(3,7,11,14)(4,8,13,16)(5,9,15,12)(17,18)(19,25)(20,30)(21,27)(22,26)(23,24)(28,32)(29,31)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,19,24,21)(18,25,23,27)(20,29,22,28)(26,32,30,31)(34,40)(35,44)(37,47)(42,46);(3,11)(4,13)(7,14)(8,16)(17,20,24,22)(18,26,23,30)(19,28,21,29)(25,31,27,32)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,21,24,19)(18,27,23,25)(20,28,22,29)(26,31,30,32)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,22,24,20)(18,26,23,30)(19,29,21,28)(25,31,27,32)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,23,24,18)(19,27,21,25)(20,26,22,30)(28,31,29,32)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815543);