#508741

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415937);