#6460049

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=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, 6367245);