#8534742

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

4142 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441938);