#428193

order	512
nilpotency class	3
isoclinism	isoclinic class of #428193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15379
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335389);