#481504

order	512
nilpotency class	4
isoclinism	isoclinic class of #481477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1043 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388700);