#7714780

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4482 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)(3,4)(5,6)(7,8)(10,13)(14,16)(18,23)(20,31)(21,34)(24,28)(29,39)(30,40)(38,45)(46,48);(9,10,12,13)(11,14,15,16)(18,23)(19,27)(25,32)(26,35)(29,39)(30,40)(38,45)(43,47);(2,6)(3,7)(17,18,22,23)(19,28,27,24)(20,32,31,25)(21,26,34,35)(29,36,39,41)(30,42,40,37)(33,45,44,38)(43,46,47,48);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,19,22,27)(18,24,23,28)(20,29,31,39)(21,30,34,40)(25,36,32,41)(26,37,35,42)(33,43,44,47)(38,46,45,48);(1,4)(2,7)(3,6)(5,8)(17,20,22,31)(18,25,23,32)(19,29,27,39)(21,33,34,44)(24,36,28,41)(26,38,35,45)(30,43,40,47)(37,46,42,48);(17,21)(18,26)(19,30)(20,33)(22,34)(23,35)(24,37)(25,38)(27,40)(28,42)(29,43)(31,44)(32,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=gh,e^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, 7621976);