#389253

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fh,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296449);