#100735

order	512
nilpotency class	3
isoclinism	isoclinic class of #100715
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

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

normal subgroups only

This group has 408 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,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, 7931);