#7715519

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×#525	C₂²
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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=h,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, 7622715);