#8818225

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

4227 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ghi,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, 8725421);