#380116

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,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^b=di,d^c=de,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287312);