#508175

order	512
nilpotency class	4
isoclinism	isoclinic class of #508024
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,26)(14,30)(15,21)(16,22)(18,29)(19,25)(23,34)(24,32)(27,35)(28,37)(31,38)(33,39)(36,40);(2,5)(6,8)(9,11)(10,15)(12,21)(13,27)(14,23)(16,26)(17,20)(18,37)(19,34)(22,24)(25,31)(28,40)(29,39)(30,38)(32,35)(33,36);(3,7)(6,8)(9,11)(10,16)(12,21)(13,28)(14,31)(15,32)(17,35)(18,38)(19,37)(20,26)(22,29)(23,25)(24,39)(27,40)(30,33)(34,36);(1,3,4,7)(2,6,5,8)(11,21)(15,20)(16,35)(22,27)(23,31)(28,39)(33,37)(34,38)

Polycyclic presentation:

2^9|d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,d^c=de,f^c=fh,h^c=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, 415371);