#128366

order	512
nilpotency class	4
isoclinism	isoclinic class of #11499
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

572 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=i,f^2=h,b^a=bd,f^a=fg,c^b=ch,d^b=dg,g^b=gi,f^d=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, 35562);