#8119633

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

4231 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026829);