#431129

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

843 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338325);