#100865

order	512
nilpotency class	3
isoclinism	isoclinic class of #100827
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

363 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=gh,e^2=ghi,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eghi,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8061);