#421635

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

971 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)(11,22)(12,26)(13,31)(14,20)(15,19)(16,21)(17,24)(18,29)(23,32)(25,33)(27,37)(28,36)(30,35)(34,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,17)(12,23)(13,14)(15,24)(16,25)(18,34)(19,33)(20,22)(21,31)(26,40)(27,28)(29,38)(30,39)(32,36)(35,37);(1,3,5,4)(2,7,6,8)(9,12,14,28,16,30,15,29)(10,18,19,35,21,36,20,26)(11,23,24,38,25,39,13,27)(17,32,22,37,31,34,33,40);(1,4,5,3)(2,7,6,8)(9,13,15,25,16,24,14,11)(10,17,20,33,21,31,19,22)(12,27,29,39,30,38,28,23)(18,32,26,40,36,34,35,37)

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=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, 328831);