#8291611

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

4245 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)(3,5)(4,12)(6,7)(8,10)(9,15)(11,16)(13,14)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,39,43)(35,40,46,47)(36,45,37,41)(38,42,48,44);(2,7)(3,10)(9,12)(11,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35,39,46)(34,40,43,47)(36,42,37,44)(38,45,48,41);(2,7)(5,8)(9,12)(14,16)(19,25)(23,26)(27,31)(30,32)(33,36,39,37)(34,41,43,45)(35,44,46,42)(38,40,48,47);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,15)(12,16)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,39,36)(34,41,43,45)(35,42,46,44)(38,40,48,47);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)(33,34)(35,40)(36,41)(37,45)(38,42)(39,43)(44,48)(46,47);(1,5,6,8)(2,3,7,10)(4,14,15,16)(9,11,12,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38)(34,42)(35,45)(36,47)(37,40)(39,48)(41,46)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198807);