#465308

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=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, 372504);