#478503

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385699);