#6458772

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365968);