#498670

order	512
nilpotency class	3
isoclinism	isoclinic class of #387433
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2372
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23273
prime radical	#532	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ch,d^b=deh,e^b=ei,g^b=gi,d^c=di,g^c=gi,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, 405866);