#102046

order	512
nilpotency class	3
isoclinism	isoclinic class of #102045
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×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

428 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(2,5)(3,6)(7,8)(9,10,14,15,13,11,12,16)(17,18)(19,23,28,37)(20,30)(21,33)(22,25)(24,31)(26,34)(27,39,40,48)(29,43,42,36)(32,44)(35,46,47,41)(38,45);(1,2,6,8,4,7,3,5)(9,11,14,16,13,10,12,15)(17,19,22,28)(18,23)(20,27,32,40)(21,29,34,42)(24,35)(25,37)(26,36)(30,39)(31,41,45,46)(33,43)(38,47)(44,48);(9,12,13,14)(10,15,11,16)(17,20,21,31,22,32,34,45)(18,24,26,30,25,38,33,44)(19,27,29,41,28,40,42,46)(23,35,36,39,37,47,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,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, 9242);