#8926692

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

3991 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833888);