#508173

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

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

normal subgroups only

This group has 879 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)(11,15)(12,16)(13,27)(14,18)(17,24)(19,25)(20,29)(21,28)(22,38)(23,32)(26,33)(30,37)(31,36)(34,40)(35,39);(2,5)(6,8)(9,11)(10,15)(12,23)(13,22)(14,20)(16,32)(17,35)(18,29)(19,31)(21,26)(24,39)(25,36)(27,38)(28,33)(30,34)(37,40);(3,7)(6,8)(10,16)(12,24)(13,25)(15,28)(17,19)(18,33)(21,37)(22,36)(23,39)(26,40)(27,34)(29,32)(30,31)(35,38);(1,3,4,7)(2,6,5,8)(11,20)(15,29)(21,23)(22,36)(28,32)(30,35)(31,38)(37,39)

Polycyclic presentation:

2^9|d^2=e,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, 415369);