#7715450

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	#38903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,25)(21,33)(22,23)(26,34)(27,35)(28,37)(29,43)(30,39)(31,45)(32,36)(38,41)(40,46)(42,48)(44,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(18,23)(19,27)(25,36)(26,33)(28,30)(29,41)(40,45)(42,44);(1,2,5,8)(3,6,4,7)(9,11,13,12)(10,14,16,15)(17,19,22,27)(18,24,23,35)(20,30,32,28)(21,29,34,41)(25,39,36,37)(26,38,33,43)(31,44,46,42)(40,48,45,47);(1,2)(3,6)(4,7)(5,8)(17,18)(19,24)(20,25)(21,26)(22,23)(27,35)(28,37)(29,38)(30,39)(31,40)(32,36)(33,34)(41,43)(42,47)(44,48)(45,46);(1,4,5,3)(2,7,8,6)(9,12,13,11)(10,15,16,14)(17,20,22,32)(18,25,23,36)(19,28,27,30)(21,31,34,46)(24,37,35,39)(26,40,33,45)(29,42,41,44)(38,47,43,48);(17,21,22,34)(18,26,23,33)(19,29,27,41)(20,31,32,46)(24,38,35,43)(25,40,36,45)(28,42,30,44)(37,47,39,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622646);