#438231

order	512
nilpotency class	3
isoclinism	isoclinic class of #438230
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15615
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

968 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=e,e^2=i,f^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,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, 345427);