#8118607

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

4214 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)(3,9)(4,6)(5,12)(7,11)(8,13)(10,14)(15,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(36,43)(40,45)(41,47)(46,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(2,4)(7,9)(8,12)(14,15)(21,29)(25,32)(27,28)(30,31)(34,38)(39,44)(40,45)(46,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,40)(35,41)(37,43)(38,45)(39,46)(42,47)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8025803);