#100704

order	512
nilpotency class	3
isoclinism	isoclinic class of #100555
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egi,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7900);