#432049

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
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

946 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)(9,10,15,20)(11,21,25,30)(12,26,29,18)(13,16,14,19)(17,23,33,24)(22,35,38,40)(27,31,28,34)(32,37,39,36);(1,2,5,6)(3,7,4,8)(10,16)(12,27)(13,14)(17,30)(18,26)(19,20)(21,33)(22,36)(23,24)(28,29)(35,40)(37,38);(1,3,5,4)(2,7,6,8)(9,11)(10,17)(12,22)(13,24)(14,23)(15,25)(16,30)(18,32)(19,21)(20,33)(26,39)(27,37)(28,36)(29,38)(31,40)(34,35);(1,4,5,3)(2,7,6,8)(9,12)(10,18)(11,22)(13,28)(14,27)(15,29)(16,31)(17,32)(19,34)(20,26)(21,35)(23,37)(24,36)(25,38)(30,40)(33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ef,d^2=ef,e^2=i,f^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,e^c=ei,f^c=fi,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, 339245);