#102047

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
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	#390
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

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

normal subgroups only

This group has 366 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^b=fh

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, 9243);