#498642

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23250
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

390 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

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, 405838);