#8295618

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8202814);