#100612

order	512
nilpotency class	3
isoclinism	isoclinic class of #100555
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

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

392 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=h,d^2=gh,e^2=gi,f^2=h,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, 7808);