#100738

order	512
nilpotency class	3
isoclinism	isoclinic class of #100715
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

380 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gh,e^2=hi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,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, 7934);