#432370

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15456
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

889 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,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, 339566);