#8174606

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081802);