#7673662

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580858);