#7674646

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

4806 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=i,f^2=i,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, 7581842);