#8924737

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

4233 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(34,36)(40,41)(42,44)(47,48);(2,7)(4,9)(8,10)(12,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,7)(3,11)(5,15)(12,14)(18,22)(20,29)(23,25)(26,32)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,12)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4,6,9)(2,5,7,15)(3,12,11,14)(8,16,10,13)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5)(2,9)(3,13)(4,7)(6,15)(8,12)(10,14)(11,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=ghi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8831933);