#98917

order	512
nilpotency class	3
isoclinism	isoclinic class of #98904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

314 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dhi,c^2=h,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,e^c=egh,f^c=fg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6113);