#126950

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

656 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,e^2=h,f^2=i,b^a=bd,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34146);