#473465

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380661);