#100768

order	512
nilpotency class	3
isoclinism	isoclinic class of #100723
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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

347 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 88 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:

(2,5)(3,7)(6,8)(9,10,13,17,15,18,14,23)(11,19,20,24,12,21,22,16)(25,26)(27,33,35,40)(28,30)(29,32)(31,34,38,37)(36,39);(1,2,3,5,4,6,7,8)(9,11,15,12)(10,16,18,24)(13,22,14,20)(17,21,23,19)(26,30)(29,36)(31,38)(34,37);(2,6)(5,8)(9,12,15,11)(10,16,18,24)(13,22,14,20)(17,19,23,21)(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=dfi,c^2=h,d^2=gh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7964);