#438332

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
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

1139 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:

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

Polycyclic presentation:

2^9|b^2=egi,c^2=f,d^2=e,f^2=i,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,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, 345528);