#128372

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₄×#99
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 534 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)(11,20,23,35,24,36,30,40)(13,25,27,39,28,18,32,33)(16,29,31,38,21,34,37,22);(1,2,3,6,4,5,7,8)(9,11)(10,16)(12,21)(13,22,28,38)(14,23)(15,24)(17,30)(18,20,25,36)(19,31)(26,37)(27,29,32,34)(33,35,39,40);(2,5)(6,8)(9,12,14,26,15,10,17,19)(11,21,23,37,24,16,30,31)(13,25,27,39,28,18,32,33)(20,34,35,22,36,29,40,38)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=i,f^2=h,b^a=bd,d^a=di,f^a=fgi,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, 35568);