#466490

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,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,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, 373686);