#6763218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6670414);