#508028

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1074 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1074 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)(11,20)(12,25)(13,14)(16,31)(17,33)(18,30)(21,36)(22,27)(23,37)(24,38)(26,39)(28,29)(32,40)(34,35);(2,5)(6,8)(9,10)(11,21)(12,26)(13,30)(14,18)(15,19)(16,22)(17,29)(20,36)(23,32)(24,35)(25,39)(27,31)(28,33)(34,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,24)(10,16,19,32)(12,27,29,37)(13,23,14,22)(17,34,26,36)(18,35,30,21)(20,28,38,25)(31,39,40,33);(3,7)(6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415224);