#101825

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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

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

normal subgroups only

This group has 364 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=i,d^2=gh,e^2=ghi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

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, 9021);