#508711

order	512
nilpotency class	4
isoclinism	isoclinic class of #508667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 467 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=f,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,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, 415907);