#100642

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

615 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ghi,d^2=gh,e^2=gi,f^2=gh,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, 7838);