#497207

order	512
nilpotency class	3
isoclinism	isoclinic class of #384259
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2333
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22339
prime radical	#3050	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,d^b=dh,f^b=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404403);