#8926691

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:

(1,2,7,8)(3,9,4,10)(5,11,16,6)(12,13,14,15)(17,18)(19,23)(20,30)(21,25)(22,26)(24,28)(27,29)(31,32)(34,38)(39,44)(40,45)(46,48);(2,8)(5,16)(9,10)(12,14)(19,27)(21,31)(23,29)(25,32)(36,43)(40,45)(41,47)(46,48);(1,3,7,4)(2,9,8,10)(5,12,16,14)(6,15,11,13)(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);(1,4,7,3)(2,10,8,9)(5,12,16,14)(6,15,11,13)(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);(1,5)(2,11)(3,12)(4,14)(6,8)(7,16)(9,13)(10,15)(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);(1,6,7,11)(2,5,8,16)(3,13,4,15)(9,14,10,12)(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|a^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, 8833887);