#442155

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	#2420
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi

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, 349351);