#7744215

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7651411);