#508119

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

762 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,24,27)(23,31,32,49)(25,37,35,50)(26,40,36,30)(28,44,42,33)(29,46,43,39)(34,51,41,47)(38,48,52,45);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(22,27)(28,42)(29,43)(30,40)(31,49)(37,50)(45,48)(47,51);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,23,24,32)(22,28,27,42)(25,38,35,52)(26,41,36,34)(29,47,43,51)(30,48,40,45)(31,50,49,37)(33,46,44,39);(8,16)(11,18)(14,20)(17,19)(23,32)(28,42)(31,49)(33,44)(34,41)(38,52)(45,48)(47,51)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^c=de,f^c=fh,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, 415315);