#375673

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1168 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 250 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)(3,7)(4,5)(6,8)(9,10,14,19,15,18,32,12)(11,20,23,21,24,16,30,25)(13,27,29,34,28,17,31,26)(22,33,38,36,37,35,39,40);(2,5)(6,7)(9,11,15,24)(10,16,18,20)(12,25,19,21)(13,22,28,37)(14,30,32,23)(17,33,27,35)(26,40,34,36)(29,39,31,38);(9,12,14,18,15,19,32,10)(11,21,23,20,24,25,30,16)(13,26,31,27,28,34,29,17)(22,36,39,35,37,40,38,33);(1,3,4,8)(2,6,5,7)(9,13)(10,17)(11,22)(12,26)(14,31)(15,28)(16,33)(18,27)(19,34)(20,35)(21,36)(23,39)(24,37)(25,40)(29,32)(30,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282869);