#128274

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×C₂²⋊C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cgi,f^a=fi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35470);