#8115493

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

4463 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)(3,5)(4,7)(6,8)(9,10,13,11)(12,15,16,14)(18,23)(20,31)(21,33)(24,28)(29,39)(30,40)(38,45)(46,48);(2,3)(6,7)(10,11)(14,15)(18,23)(19,27)(20,31)(26,35)(30,40)(32,44)(36,41)(46,48);(1,2,5,3)(4,6,8,7)(9,11,13,10)(12,15,16,14)(17,18)(19,28)(20,25)(21,26)(22,23)(24,27)(29,41)(30,42)(31,34)(32,38)(33,35)(36,39)(37,40)(43,48)(44,45)(46,47);(1,3,5,2)(4,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,19,22,27)(18,24,23,28)(20,29,31,39)(21,30,33,40)(25,36,34,41)(26,37,35,42)(32,43,44,47)(38,46,45,48);(1,4,5,8)(2,6,3,7)(17,20,22,31)(18,25,23,34)(19,29,27,39)(21,32,33,44)(24,36,28,41)(26,38,35,45)(30,43,40,47)(37,46,42,48);(17,21,22,33)(18,26,23,35)(19,30,27,40)(20,32,31,44)(24,37,28,42)(25,38,34,45)(29,43,39,47)(36,46,41,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^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, 8022689);