#428405

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1087 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,g^d=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, 335601);