#380571

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287767);