#129052

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

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

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, 36248);