#8527992

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

4108 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(2,7)(3,9)(10,15)(11,13)(18,24)(19,20)(26,29)(27,28)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(2,7)(4,8)(10,15)(12,16)(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35,39,36)(34,41,37,40)(38,43,46,44)(42,48,45,47);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,36,39,35)(34,41,37,40)(38,43,46,44)(42,47,45,48);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=g,e^2=g,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, 8435188);