#348065

order	512
nilpotency class	3
isoclinism	isoclinic class of #348058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3483	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

893 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 893 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255261);