#432052

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339248);