#477251

order	512
nilpotency class	3
isoclinism	isoclinic class of #476109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 894 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 384447);