#467357

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,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, 374553);