#432051

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
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

949 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)(11,21,25,31)(12,26,30,35)(13,19,14,17)(16,24,33,23)(18,28,36,29)(22,32,39,40)(27,37,38,34);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,31)(18,34)(19,21)(20,33)(22,28)(26,40)(29,39)(30,38)(32,35)(36,37);(1,3,5,4)(2,7,6,8)(10,17)(13,14)(16,31)(18,26)(19,20)(21,33)(23,24)(27,38)(28,29)(32,34)(35,36)(37,40);(1,4,5,3)(2,7,6,8)(9,12)(10,18)(11,22)(13,29)(14,28)(15,30)(16,32)(17,26)(19,35)(20,36)(21,37)(23,38)(24,27)(25,39)(31,34)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ef,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,h^a=hi,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,h^d=hi

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, 339247);