#100799

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, 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

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

normal subgroups only

This group has 335 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)(9,10,13,16,14,18,12,17)(11,19,21,23,22,24,20,15)(25,26,31,34,30,27,29,33)(28,32,38,40,37,35,36,39);(1,2,3,5,4,6,7,8)(9,11)(10,15,18,23)(12,20)(13,21)(14,22)(16,19,17,24)(25,27,30,26)(28,35,37,32)(29,33,31,34)(36,39,38,40);(10,16)(12,13)(15,19)(17,18)(20,21)(23,24)(25,28,30,37)(26,32,27,35)(29,38,31,36)(33,40,34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=gh,e^2=hi,f^2=i,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, 7995);