#401983

order	512
nilpotency class	3
isoclinism	isoclinic class of #401983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²⋊#401
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

971 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)(3,12,14,7)(4,8,17,25)(5,9)(6,10)(11,19)(13,23)(15,22,28,21)(16,20)(18,24,30,26)(27,32)(29,31);(1,3)(2,7)(4,16,17,13)(5,15)(6,14)(8,23,25,20)(9,21)(10,12)(11,22)(18,29,30,27)(19,28)(24,32,26,31);(1,4)(2,8,10,25)(3,13)(5,18)(6,17)(7,20,12,23)(9,24,11,26)(14,16)(15,27)(19,30)(21,31,22,32)(28,29);(2,9)(5,19)(7,21)(8,24)(10,11)(12,22)(15,28)(18,30)(20,31)(23,32)(25,26)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309179);