#101503

order	512
nilpotency class	3
isoclinism	isoclinic class of #101343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

433 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gh,e^2=h,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8699);