#8818125

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	#41210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725321);