#8115393

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
prime radical	#34641	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4896 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022589);