#441801

order	512
nilpotency class	3
isoclinism	isoclinic class of #441801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15695
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 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)(3,8,15,27)(4,17,19,32)(5,20)(6,12)(7,11)(9,14,25,21)(10,13)(16,26,23,29)(18,30)(22,24)(28,31);(1,3,7,23,6,15,18,16)(2,8,11,29,12,27,30,26)(4,5,21,22,19,13,14,31)(9,24,32,10,25,28,17,20);(1,4,18,21,6,19,7,14)(2,9,11,17,12,25,30,32)(3,13,23,22,15,5,16,31)(8,24,26,10,27,28,29,20);(1,5,6,13)(2,10)(3,14,15,21)(4,16,19,23)(7,22,18,31)(8,25)(9,27)(11,28)(12,20)(17,29)(24,30)(26,32)

Polycyclic presentation:

2^9|a^2=eg,b^2=efg,c^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 348997);