#477172

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1088 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,4)(3,6,7,8)(9,10,14,19,15,20,31,38)(11,16,21,32,22,24,39,40)(12,23,25,33,26,36,29,17)(13,18,27,34,28,35,30,37);(2,4)(6,8)(9,11,14,21,15,22,31,39)(10,16,19,32,20,24,38,40)(12,18,25,34,26,35,29,37)(13,23,27,33,28,36,30,17);(1,3,5,7)(2,6,4,8)(9,12,15,26)(10,17,20,33)(11,18,22,35)(13,24,28,16)(14,29,31,25)(19,36,38,23)(21,37,39,34)(27,32,30,40);(1,4,5,2)(3,6,7,8)(9,13,15,28)(10,18,20,35)(11,17,22,33)(12,24,26,16)(14,30,31,27)(19,37,38,34)(21,36,39,23)(25,32,29,40)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=ei,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384368);