#445016

order	512
nilpotency class	3
isoclinism	isoclinic class of #445016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,f^d=fi,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, 352212);