#436094

order	512
nilpotency class	3
isoclinism	isoclinic class of #421447
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2474
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15549
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

992 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,d^2=f,g^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^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343290);