#8755068

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662264);