#101442

order	512
nilpotency class	3
isoclinism	isoclinic class of #101343
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

371 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gh,d^2=gh,e^2=h,f^2=i,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, 8638);