#430408

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1197 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)(5,6,12,16,11,7,10,15)(8,17,23,19,22,13,21,30)(9,24,28,14,27,25,26,18)(20,33,36,31,35,32,34,29);(1,3,4,2)(5,7,11,6)(8,19,22,30)(9,25,27,24)(10,15,12,16)(13,23,17,21)(14,26,18,28)(20,31,35,29)(32,36,33,34);(2,3)(5,8)(6,13)(7,17)(9,20)(10,21)(11,22)(12,23)(14,29)(15,19)(16,30)(18,31)(24,32)(25,33)(26,34)(27,35)(28,36);(2,3)(5,9,12,28,11,27,10,26)(6,14,16,25,7,18,15,24)(8,20,23,36,22,35,21,34)(13,29,30,33,17,31,19,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 337604);